Properties

Label 4-18e4-1.1-c2e2-0-9
Degree $4$
Conductor $104976$
Sign $1$
Analytic cond. $77.9399$
Root an. cond. $2.97125$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 12·4-s − 7·5-s − 15·7-s + 32·8-s − 28·10-s + 15·11-s − 20·13-s − 60·14-s + 80·16-s − 16·17-s − 84·20-s + 60·22-s + 6·23-s + 25·25-s − 80·26-s − 180·28-s − 10·29-s + 93·31-s + 192·32-s − 64·34-s + 105·35-s − 20·37-s − 224·40-s + 50·41-s + 30·43-s + 180·44-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s − 7/5·5-s − 2.14·7-s + 4·8-s − 2.79·10-s + 1.36·11-s − 1.53·13-s − 4.28·14-s + 5·16-s − 0.941·17-s − 4.19·20-s + 2.72·22-s + 6/23·23-s + 25-s − 3.07·26-s − 6.42·28-s − 0.344·29-s + 3·31-s + 6·32-s − 1.88·34-s + 3·35-s − 0.540·37-s − 5.59·40-s + 1.21·41-s + 0.697·43-s + 4.09·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(104976\)    =    \(2^{4} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(77.9399\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{324} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 104976,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(5.034098682\)
\(L(\frac12)\) \(\approx\) \(5.034098682\)
\(L(2)\) 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$C_1$ \( ( 1 - p T )^{2} \)
3 \( 1 \)
good5$C_2^2$ \( 1 + 7 T + 24 T^{2} + 7 p^{2} T^{3} + p^{4} T^{4} \)
7$C_2$ \( ( 1 + 2 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \)
11$C_2^2$ \( 1 - 15 T + 196 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} \)
13$C_2^2$ \( 1 + 20 T + 231 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} \)
17$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 614 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 541 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2^2$ \( 1 + 10 T - 741 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} \)
31$C_1$$\times$$C_2$ \( ( 1 - p T )^{2}( 1 - p T + p^{2} T^{2} ) \)
37$C_2$ \( ( 1 + 10 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 50 T + 819 T^{2} - 50 p^{2} T^{3} + p^{4} T^{4} \)
43$C_2^2$ \( 1 - 30 T + 2149 T^{2} - 30 p^{2} T^{3} + p^{4} T^{4} \)
47$C_2^2$ \( 1 - 150 T + 9709 T^{2} - 150 p^{2} T^{3} + p^{4} T^{4} \)
53$C_2$ \( ( 1 + 47 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 60 T + 4681 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} \)
61$C_2^2$ \( 1 - 64 T + 375 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} \)
67$C_2^2$ \( 1 + 150 T + 11989 T^{2} + 150 p^{2} T^{3} + p^{4} T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 + 55 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 12 T + 6289 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 51 T + 7756 T^{2} - 51 p^{2} T^{3} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 25 T - 8784 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} \)
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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

−11.93791441177328006047070624551, −11.44977644017610314374418431271, −10.98790196049466791019023076800, −10.30443053726449578876026366877, −10.03853362642129220429886446092, −9.371552485020353111961936347218, −8.866651042766795223212737825671, −8.047840781064381635721846476079, −7.31506890280117228253003539996, −7.17618434184629046055575624431, −6.71608669788890479124141839745, −6.11134121490630173293614953523, −5.94553230619523667214420998667, −4.83253112275420342628412190584, −4.32689636327937345489990999091, −4.12285023260239010384254145526, −3.38643428487389207944591234254, −2.88207354318626171196787070248, −2.38927377161762216889466560243, −0.76111421022900318930667311233, 0.76111421022900318930667311233, 2.38927377161762216889466560243, 2.88207354318626171196787070248, 3.38643428487389207944591234254, 4.12285023260239010384254145526, 4.32689636327937345489990999091, 4.83253112275420342628412190584, 5.94553230619523667214420998667, 6.11134121490630173293614953523, 6.71608669788890479124141839745, 7.17618434184629046055575624431, 7.31506890280117228253003539996, 8.047840781064381635721846476079, 8.866651042766795223212737825671, 9.371552485020353111961936347218, 10.03853362642129220429886446092, 10.30443053726449578876026366877, 10.98790196049466791019023076800, 11.44977644017610314374418431271, 11.93791441177328006047070624551

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