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Properties

Label 4-90e2-1.1-c17e2-0-5
Degree $4$
Conductor $8100$
Sign $1$
Analytic cond. $27191.9$
Root an. cond. $12.8413$
Motivic weight $17$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 512·2-s + 1.96e5·4-s + 7.81e5·5-s + 2.76e7·7-s − 6.71e7·8-s − 4.00e8·10-s − 6.45e7·11-s + 2.89e9·13-s − 1.41e10·14-s + 2.14e10·16-s − 1.58e9·17-s + 4.42e10·19-s + 1.53e11·20-s + 3.30e10·22-s − 4.87e11·23-s + 4.57e11·25-s − 1.48e12·26-s + 5.44e12·28-s − 3.98e12·29-s − 5.49e12·31-s − 6.59e12·32-s + 8.09e11·34-s + 2.16e13·35-s − 6.27e13·37-s − 2.26e13·38-s − 5.24e13·40-s − 2.34e13·41-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.894·5-s + 1.81·7-s − 1.41·8-s − 1.26·10-s − 0.0907·11-s + 0.984·13-s − 2.56·14-s + 5/4·16-s − 0.0549·17-s + 0.597·19-s + 1.34·20-s + 0.128·22-s − 1.29·23-s + 3/5·25-s − 1.39·26-s + 2.72·28-s − 1.48·29-s − 1.15·31-s − 1.06·32-s + 0.0776·34-s + 1.62·35-s − 2.93·37-s − 0.844·38-s − 1.26·40-s − 0.457·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(27191.9\)
Root analytic conductor: \(12.8413\)
Motivic weight: \(17\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 8100,\ (\ :17/2, 17/2),\ 1)\)

Particular Values

\(L(9)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{19}{2})\) 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$C_1$ \( ( 1 + p^{8} T )^{2} \)
3 \( 1 \)
5$C_1$ \( ( 1 - p^{8} T )^{2} \)
good7$D_{4}$ \( 1 - 27684196 T + 93506538073374 p T^{2} - 27684196 p^{17} T^{3} + p^{34} T^{4} \)
11$D_{4}$ \( 1 + 533184 p^{2} T - 3200366345168954 p^{2} T^{2} + 533184 p^{19} T^{3} + p^{34} T^{4} \)
13$D_{4}$ \( 1 - 2895838468 T + 1418824834264307094 p T^{2} - 2895838468 p^{17} T^{3} + p^{34} T^{4} \)
17$D_{4}$ \( 1 + 1580212596 T + \)\(13\!\cdots\!58\)\( T^{2} + 1580212596 p^{17} T^{3} + p^{34} T^{4} \)
19$D_{4}$ \( 1 - 44213712760 T + \)\(27\!\cdots\!78\)\( T^{2} - 44213712760 p^{17} T^{3} + p^{34} T^{4} \)
23$D_{4}$ \( 1 + 487549782828 T + \)\(24\!\cdots\!02\)\( T^{2} + 487549782828 p^{17} T^{3} + p^{34} T^{4} \)
29$D_{4}$ \( 1 + 3987314863500 T + \)\(18\!\cdots\!18\)\( T^{2} + 3987314863500 p^{17} T^{3} + p^{34} T^{4} \)
31$D_{4}$ \( 1 + 5492261339336 T + \)\(52\!\cdots\!46\)\( T^{2} + 5492261339336 p^{17} T^{3} + p^{34} T^{4} \)
37$D_{4}$ \( 1 + 62715287637884 T + \)\(18\!\cdots\!98\)\( T^{2} + 62715287637884 p^{17} T^{3} + p^{34} T^{4} \)
41$D_{4}$ \( 1 + 23411477277324 T + \)\(10\!\cdots\!06\)\( T^{2} + 23411477277324 p^{17} T^{3} + p^{34} T^{4} \)
43$D_{4}$ \( 1 + 124856923191092 T + \)\(14\!\cdots\!02\)\( T^{2} + 124856923191092 p^{17} T^{3} + p^{34} T^{4} \)
47$D_{4}$ \( 1 - 185946612123564 T + \)\(57\!\cdots\!98\)\( T^{2} - 185946612123564 p^{17} T^{3} + p^{34} T^{4} \)
53$D_{4}$ \( 1 + 359339780647668 T + \)\(33\!\cdots\!82\)\( T^{2} + 359339780647668 p^{17} T^{3} + p^{34} T^{4} \)
59$D_{4}$ \( 1 + 902179170360600 T + \)\(20\!\cdots\!38\)\( T^{2} + 902179170360600 p^{17} T^{3} + p^{34} T^{4} \)
61$D_{4}$ \( 1 + 1564422918967676 T + \)\(36\!\cdots\!86\)\( T^{2} + 1564422918967676 p^{17} T^{3} + p^{34} T^{4} \)
67$D_{4}$ \( 1 + 5839738931054684 T + \)\(30\!\cdots\!18\)\( T^{2} + 5839738931054684 p^{17} T^{3} + p^{34} T^{4} \)
71$D_{4}$ \( 1 - 67588560434136 T + \)\(50\!\cdots\!06\)\( T^{2} - 67588560434136 p^{17} T^{3} + p^{34} T^{4} \)
73$D_{4}$ \( 1 - 3533390699585668 T + \)\(90\!\cdots\!62\)\( T^{2} - 3533390699585668 p^{17} T^{3} + p^{34} T^{4} \)
79$D_{4}$ \( 1 - 19002656396552080 T + \)\(43\!\cdots\!18\)\( T^{2} - 19002656396552080 p^{17} T^{3} + p^{34} T^{4} \)
83$D_{4}$ \( 1 + 261145638254436 p T + \)\(75\!\cdots\!82\)\( T^{2} + 261145638254436 p^{18} T^{3} + p^{34} T^{4} \)
89$D_{4}$ \( 1 - 97499522192222220 T + \)\(48\!\cdots\!58\)\( T^{2} - 97499522192222220 p^{17} T^{3} + p^{34} T^{4} \)
97$D_{4}$ \( 1 - 99889937855386756 T + \)\(14\!\cdots\!58\)\( T^{2} - 99889937855386756 p^{17} T^{3} + p^{34} T^{4} \)
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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

−10.34360703390788203190269984966, −10.33377234436136465954314807016, −9.327916335656005432639207920140, −9.066516788035593427385432067389, −8.514181838901311894448142536526, −8.057500405514226981069234832560, −7.53809528707891868912732312089, −7.11109672840241981022014342549, −6.23719228170987842081856182375, −5.89066041128840889236808566856, −5.13035499266723985691186067510, −4.86177587349639523912222751992, −3.60355241164618189742303043783, −3.41064448306954156659140820012, −2.18039537382731144699957228275, −1.91659506850292710975752477745, −1.45526795872369307561004690532, −1.25496525249032558828125316522, 0, 0, 1.25496525249032558828125316522, 1.45526795872369307561004690532, 1.91659506850292710975752477745, 2.18039537382731144699957228275, 3.41064448306954156659140820012, 3.60355241164618189742303043783, 4.86177587349639523912222751992, 5.13035499266723985691186067510, 5.89066041128840889236808566856, 6.23719228170987842081856182375, 7.11109672840241981022014342549, 7.53809528707891868912732312089, 8.057500405514226981069234832560, 8.514181838901311894448142536526, 9.066516788035593427385432067389, 9.327916335656005432639207920140, 10.33377234436136465954314807016, 10.34360703390788203190269984966

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