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Properties

Label 4-2e2-1.1-c43e2-0-1
Degree $4$
Conductor $4$
Sign $1$
Analytic cond. $548.593$
Root an. cond. $4.83963$
Motivic weight $43$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.19e6·2-s − 2.23e10·3-s + 1.31e13·4-s − 4.73e13·5-s − 9.37e16·6-s − 2.22e17·7-s + 3.68e19·8-s + 1.92e20·9-s − 1.98e20·10-s − 4.18e22·11-s − 2.94e23·12-s − 1.51e24·13-s − 9.33e23·14-s + 1.05e24·15-s + 9.67e25·16-s − 6.73e25·17-s + 8.07e26·18-s + 1.23e27·19-s − 6.24e26·20-s + 4.97e27·21-s − 1.75e29·22-s + 3.81e28·23-s − 8.24e29·24-s − 2.09e30·25-s − 6.33e30·26-s − 4.78e30·27-s − 2.93e30·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.23·3-s + 3/2·4-s − 0.0443·5-s − 1.74·6-s − 0.150·7-s + 1.41·8-s + 0.586·9-s − 0.0627·10-s − 1.70·11-s − 1.84·12-s − 1.69·13-s − 0.212·14-s + 0.0547·15-s + 5/4·16-s − 0.236·17-s + 0.829·18-s + 0.395·19-s − 0.0665·20-s + 0.185·21-s − 2.41·22-s + 0.201·23-s − 1.74·24-s − 1.84·25-s − 2.39·26-s − 0.805·27-s − 0.225·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+43/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $1$
Analytic conductor: \(548.593\)
Root analytic conductor: \(4.83963\)
Motivic weight: \(43\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4,\ (\ :43/2, 43/2),\ 1)\)

Particular Values

\(L(22)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{45}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - p^{21} T )^{2} \)
good3$D_{4}$ \( 1 + 827467928 p^{3} T + 192265902511906 p^{13} T^{2} + 827467928 p^{46} T^{3} + p^{86} T^{4} \)
5$D_{4}$ \( 1 + 9464131679604 p T + \)\(26\!\cdots\!26\)\( p^{7} T^{2} + 9464131679604 p^{44} T^{3} + p^{86} T^{4} \)
7$D_{4}$ \( 1 + 31780018763862704 p T + \)\(29\!\cdots\!82\)\( p^{4} T^{2} + 31780018763862704 p^{44} T^{3} + p^{86} T^{4} \)
11$D_{4}$ \( 1 + \)\(38\!\cdots\!56\)\( p T + \)\(10\!\cdots\!86\)\( p^{4} T^{2} + \)\(38\!\cdots\!56\)\( p^{44} T^{3} + p^{86} T^{4} \)
13$D_{4}$ \( 1 + \)\(15\!\cdots\!96\)\( T + \)\(12\!\cdots\!42\)\( p^{2} T^{2} + \)\(15\!\cdots\!96\)\( p^{43} T^{3} + p^{86} T^{4} \)
17$D_{4}$ \( 1 + \)\(39\!\cdots\!44\)\( p T + \)\(14\!\cdots\!54\)\( p^{3} T^{2} + \)\(39\!\cdots\!44\)\( p^{44} T^{3} + p^{86} T^{4} \)
19$D_{4}$ \( 1 - \)\(64\!\cdots\!60\)\( p T + \)\(28\!\cdots\!02\)\( p^{3} T^{2} - \)\(64\!\cdots\!60\)\( p^{44} T^{3} + p^{86} T^{4} \)
23$D_{4}$ \( 1 - \)\(38\!\cdots\!24\)\( T + \)\(93\!\cdots\!86\)\( p T^{2} - \)\(38\!\cdots\!24\)\( p^{43} T^{3} + p^{86} T^{4} \)
29$D_{4}$ \( 1 + \)\(72\!\cdots\!00\)\( T + \)\(30\!\cdots\!58\)\( p^{2} T^{2} + \)\(72\!\cdots\!00\)\( p^{43} T^{3} + p^{86} T^{4} \)
31$D_{4}$ \( 1 - \)\(40\!\cdots\!24\)\( T + \)\(40\!\cdots\!46\)\( p T^{2} - \)\(40\!\cdots\!24\)\( p^{43} T^{3} + p^{86} T^{4} \)
37$D_{4}$ \( 1 - \)\(33\!\cdots\!52\)\( T + \)\(36\!\cdots\!86\)\( p T^{2} - \)\(33\!\cdots\!52\)\( p^{43} T^{3} + p^{86} T^{4} \)
41$D_{4}$ \( 1 + \)\(14\!\cdots\!56\)\( p T + \)\(28\!\cdots\!66\)\( p^{2} T^{2} + \)\(14\!\cdots\!56\)\( p^{44} T^{3} + p^{86} T^{4} \)
43$D_{4}$ \( 1 + \)\(28\!\cdots\!56\)\( T + \)\(53\!\cdots\!98\)\( T^{2} + \)\(28\!\cdots\!56\)\( p^{43} T^{3} + p^{86} T^{4} \)
47$D_{4}$ \( 1 - \)\(18\!\cdots\!72\)\( T + \)\(20\!\cdots\!42\)\( T^{2} - \)\(18\!\cdots\!72\)\( p^{43} T^{3} + p^{86} T^{4} \)
53$D_{4}$ \( 1 - \)\(13\!\cdots\!64\)\( T + \)\(32\!\cdots\!78\)\( T^{2} - \)\(13\!\cdots\!64\)\( p^{43} T^{3} + p^{86} T^{4} \)
59$D_{4}$ \( 1 - \)\(65\!\cdots\!00\)\( T + \)\(26\!\cdots\!58\)\( T^{2} - \)\(65\!\cdots\!00\)\( p^{43} T^{3} + p^{86} T^{4} \)
61$D_{4}$ \( 1 - \)\(84\!\cdots\!44\)\( T + \)\(10\!\cdots\!46\)\( T^{2} - \)\(84\!\cdots\!44\)\( p^{43} T^{3} + p^{86} T^{4} \)
67$D_{4}$ \( 1 + \)\(38\!\cdots\!08\)\( T + \)\(10\!\cdots\!42\)\( T^{2} + \)\(38\!\cdots\!08\)\( p^{43} T^{3} + p^{86} T^{4} \)
71$D_{4}$ \( 1 + \)\(45\!\cdots\!96\)\( T + \)\(76\!\cdots\!26\)\( T^{2} + \)\(45\!\cdots\!96\)\( p^{43} T^{3} + p^{86} T^{4} \)
73$D_{4}$ \( 1 + \)\(10\!\cdots\!56\)\( T + \)\(20\!\cdots\!18\)\( T^{2} + \)\(10\!\cdots\!56\)\( p^{43} T^{3} + p^{86} T^{4} \)
79$D_{4}$ \( 1 - \)\(10\!\cdots\!20\)\( T + \)\(10\!\cdots\!78\)\( T^{2} - \)\(10\!\cdots\!20\)\( p^{43} T^{3} + p^{86} T^{4} \)
83$D_{4}$ \( 1 + \)\(99\!\cdots\!56\)\( T + \)\(45\!\cdots\!58\)\( T^{2} + \)\(99\!\cdots\!56\)\( p^{43} T^{3} + p^{86} T^{4} \)
89$D_{4}$ \( 1 - \)\(17\!\cdots\!20\)\( T + \)\(11\!\cdots\!38\)\( T^{2} - \)\(17\!\cdots\!20\)\( p^{43} T^{3} + p^{86} T^{4} \)
97$D_{4}$ \( 1 + \)\(36\!\cdots\!88\)\( T - \)\(10\!\cdots\!18\)\( T^{2} + \)\(36\!\cdots\!88\)\( p^{43} T^{3} + p^{86} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.05310831717640847041831092823, −16.53781721056229382766065810149, −15.35850698157405772267829422091, −15.03833840870441340944184556413, −13.49444893533522421817150316312, −13.10214823697675250405292929622, −11.98324779812143640646026337515, −11.59140817763000318065709372324, −10.55575780948299638664076831589, −9.766156333591056102206689597675, −7.68880914418365397749002701642, −7.20266994926007741624699724485, −5.83719755484291900652658897519, −5.45269944989909920854280068663, −4.78468723426506706567837544926, −3.71575880151204210972812025857, −2.57961558149180729557837707894, −1.85459738274636160704058680932, 0, 0, 1.85459738274636160704058680932, 2.57961558149180729557837707894, 3.71575880151204210972812025857, 4.78468723426506706567837544926, 5.45269944989909920854280068663, 5.83719755484291900652658897519, 7.20266994926007741624699724485, 7.68880914418365397749002701642, 9.766156333591056102206689597675, 10.55575780948299638664076831589, 11.59140817763000318065709372324, 11.98324779812143640646026337515, 13.10214823697675250405292929622, 13.49444893533522421817150316312, 15.03833840870441340944184556413, 15.35850698157405772267829422091, 16.53781721056229382766065810149, 17.05310831717640847041831092823

Graph of the $Z$-function along the critical line