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Properties

Label 4-10e2-1.1-c25e2-0-2
Degree $4$
Conductor $100$
Sign $1$
Analytic cond. $1568.13$
Root an. cond. $6.29282$
Motivic weight $25$
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  − 8.19e3·2-s + 5.45e5·3-s + 5.03e7·4-s − 4.88e8·5-s − 4.46e9·6-s + 3.85e10·7-s − 2.74e11·8-s − 5.21e11·9-s + 4.00e12·10-s − 8.37e12·11-s + 2.74e13·12-s − 1.49e14·13-s − 3.15e14·14-s − 2.66e14·15-s + 1.40e15·16-s + 3.35e15·17-s + 4.27e15·18-s + 3.48e15·19-s − 2.45e16·20-s + 2.10e16·21-s + 6.86e16·22-s + 7.23e16·23-s − 1.49e17·24-s + 1.78e17·25-s + 1.22e18·26-s − 2.69e17·27-s + 1.94e18·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.592·3-s + 3/2·4-s − 0.894·5-s − 0.837·6-s + 1.05·7-s − 1.41·8-s − 0.616·9-s + 1.26·10-s − 0.804·11-s + 0.888·12-s − 1.77·13-s − 1.48·14-s − 0.529·15-s + 5/4·16-s + 1.39·17-s + 0.871·18-s + 0.361·19-s − 1.34·20-s + 0.623·21-s + 1.13·22-s + 0.688·23-s − 0.837·24-s + 3/5·25-s + 2.51·26-s − 0.345·27-s + 1.57·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+25/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(1568.13\)
Root analytic conductor: \(6.29282\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 100,\ (\ :25/2, 25/2),\ 1)\)

Particular Values

\(L(13)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{27}{2})\) 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$C_1$ \( ( 1 + p^{12} T )^{2} \)
5$C_1$ \( ( 1 + p^{12} T )^{2} \)
good3$D_{4}$ \( 1 - 545212 T + 30341164286 p^{3} T^{2} - 545212 p^{25} T^{3} + p^{50} T^{4} \)
7$D_{4}$ \( 1 - 5509693852 p T + 1105473550077933138 p^{4} T^{2} - 5509693852 p^{26} T^{3} + p^{50} T^{4} \)
11$D_{4}$ \( 1 + 8379169876416 T + \)\(95\!\cdots\!46\)\( p^{2} T^{2} + 8379169876416 p^{25} T^{3} + p^{50} T^{4} \)
13$D_{4}$ \( 1 + 11495672092076 p T + \)\(87\!\cdots\!38\)\( p^{2} T^{2} + 11495672092076 p^{26} T^{3} + p^{50} T^{4} \)
17$D_{4}$ \( 1 - 3359254299255924 T + \)\(83\!\cdots\!74\)\( p T^{2} - 3359254299255924 p^{25} T^{3} + p^{50} T^{4} \)
19$D_{4}$ \( 1 - 183655863992680 p T + \)\(15\!\cdots\!18\)\( p^{2} T^{2} - 183655863992680 p^{26} T^{3} + p^{50} T^{4} \)
23$D_{4}$ \( 1 - 3146530518609684 p T + \)\(36\!\cdots\!98\)\( p^{2} T^{2} - 3146530518609684 p^{26} T^{3} + p^{50} T^{4} \)
29$D_{4}$ \( 1 + 537443787959856180 T + \)\(60\!\cdots\!98\)\( T^{2} + 537443787959856180 p^{25} T^{3} + p^{50} T^{4} \)
31$D_{4}$ \( 1 + 3624027784441312136 T + \)\(39\!\cdots\!26\)\( T^{2} + 3624027784441312136 p^{25} T^{3} + p^{50} T^{4} \)
37$D_{4}$ \( 1 - 514556017957936372 p T + \)\(16\!\cdots\!38\)\( T^{2} - 514556017957936372 p^{26} T^{3} + p^{50} T^{4} \)
41$D_{4}$ \( 1 + \)\(21\!\cdots\!56\)\( T + \)\(52\!\cdots\!86\)\( T^{2} + \)\(21\!\cdots\!56\)\( p^{25} T^{3} + p^{50} T^{4} \)
43$D_{4}$ \( 1 + 13668418927657846868 T + \)\(12\!\cdots\!42\)\( T^{2} + 13668418927657846868 p^{25} T^{3} + p^{50} T^{4} \)
47$D_{4}$ \( 1 + \)\(16\!\cdots\!56\)\( T + \)\(19\!\cdots\!98\)\( T^{2} + \)\(16\!\cdots\!56\)\( p^{25} T^{3} + p^{50} T^{4} \)
53$D_{4}$ \( 1 + \)\(99\!\cdots\!48\)\( T + \)\(47\!\cdots\!62\)\( T^{2} + \)\(99\!\cdots\!48\)\( p^{25} T^{3} + p^{50} T^{4} \)
59$D_{4}$ \( 1 + \)\(13\!\cdots\!60\)\( T + \)\(39\!\cdots\!98\)\( T^{2} + \)\(13\!\cdots\!60\)\( p^{25} T^{3} + p^{50} T^{4} \)
61$D_{4}$ \( 1 - \)\(11\!\cdots\!24\)\( T - \)\(13\!\cdots\!54\)\( T^{2} - \)\(11\!\cdots\!24\)\( p^{25} T^{3} + p^{50} T^{4} \)
67$D_{4}$ \( 1 - \)\(95\!\cdots\!44\)\( T + \)\(11\!\cdots\!98\)\( T^{2} - \)\(95\!\cdots\!44\)\( p^{25} T^{3} + p^{50} T^{4} \)
71$D_{4}$ \( 1 - \)\(83\!\cdots\!24\)\( T + \)\(34\!\cdots\!46\)\( T^{2} - \)\(83\!\cdots\!24\)\( p^{25} T^{3} + p^{50} T^{4} \)
73$D_{4}$ \( 1 - \)\(19\!\cdots\!72\)\( T + \)\(43\!\cdots\!82\)\( T^{2} - \)\(19\!\cdots\!72\)\( p^{25} T^{3} + p^{50} T^{4} \)
79$D_{4}$ \( 1 - \)\(10\!\cdots\!40\)\( T + \)\(51\!\cdots\!98\)\( T^{2} - \)\(10\!\cdots\!40\)\( p^{25} T^{3} + p^{50} T^{4} \)
83$D_{4}$ \( 1 + \)\(59\!\cdots\!48\)\( T + \)\(59\!\cdots\!62\)\( T^{2} + \)\(59\!\cdots\!48\)\( p^{25} T^{3} + p^{50} T^{4} \)
89$D_{4}$ \( 1 + \)\(18\!\cdots\!20\)\( T + \)\(11\!\cdots\!98\)\( T^{2} + \)\(18\!\cdots\!20\)\( p^{25} T^{3} + p^{50} T^{4} \)
97$D_{4}$ \( 1 + \)\(11\!\cdots\!36\)\( T + \)\(12\!\cdots\!38\)\( T^{2} + \)\(11\!\cdots\!36\)\( p^{25} T^{3} + p^{50} T^{4} \)
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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

−14.57160202439533757262151841499, −14.30587599613203452273981982122, −12.76376272715006114510808486289, −12.14663262218137303556212539135, −11.32600843163379472079279165960, −10.98159554784362422401248329837, −9.832588671207865702563175028514, −9.492537646922989460941385155021, −8.257850168107581819790151845900, −8.062759499573733997825898143470, −7.58484558598131789183616466707, −6.76482799512350825123723644684, −5.30279706047964826348639516612, −4.86923100680976881261401695389, −3.28026065029470894403981133412, −2.92572635266540744950986274490, −1.94108107346812939052426865663, −1.25014930358597200731379863363, 0, 0, 1.25014930358597200731379863363, 1.94108107346812939052426865663, 2.92572635266540744950986274490, 3.28026065029470894403981133412, 4.86923100680976881261401695389, 5.30279706047964826348639516612, 6.76482799512350825123723644684, 7.58484558598131789183616466707, 8.062759499573733997825898143470, 8.257850168107581819790151845900, 9.492537646922989460941385155021, 9.832588671207865702563175028514, 10.98159554784362422401248329837, 11.32600843163379472079279165960, 12.14663262218137303556212539135, 12.76376272715006114510808486289, 14.30587599613203452273981982122, 14.57160202439533757262151841499

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