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Properties

Label 2-38808-1.1-c1-0-30
Degree $2$
Conductor $38808$
Sign $-1$
Analytic cond. $309.883$
Root an. cond. $17.6035$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 11-s − 6·17-s − 4·19-s − 23-s + 4·25-s + 8·29-s + 7·31-s − 37-s + 4·41-s + 6·43-s − 8·47-s − 2·53-s − 3·55-s − 59-s − 4·61-s − 5·67-s − 3·71-s − 16·73-s + 2·79-s − 2·83-s + 18·85-s + 15·89-s + 12·95-s + 7·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.301·11-s − 1.45·17-s − 0.917·19-s − 0.208·23-s + 4/5·25-s + 1.48·29-s + 1.25·31-s − 0.164·37-s + 0.624·41-s + 0.914·43-s − 1.16·47-s − 0.274·53-s − 0.404·55-s − 0.130·59-s − 0.512·61-s − 0.610·67-s − 0.356·71-s − 1.87·73-s + 0.225·79-s − 0.219·83-s + 1.95·85-s + 1.58·89-s + 1.23·95-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38808\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(309.883\)
Root analytic conductor: \(17.6035\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 38808,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good5 \( 1 + 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.06129622412424, −14.73831272661168, −14.06331790052581, −13.49467581731726, −12.97485719960188, −12.36115708318212, −11.90570232390822, −11.51205887020577, −10.91060511901628, −10.49605814627023, −9.856421753472765, −9.013599181004549, −8.672969558257914, −8.134192107990830, −7.637809430971641, −6.962229552790276, −6.426450353114155, −6.015832558593863, −4.870199584002517, −4.401583953568072, −4.167431440352684, −3.243962811013039, −2.673078998450018, −1.832662945551155, −0.7963068258322876, 0, 0.7963068258322876, 1.832662945551155, 2.673078998450018, 3.243962811013039, 4.167431440352684, 4.401583953568072, 4.870199584002517, 6.015832558593863, 6.426450353114155, 6.962229552790276, 7.637809430971641, 8.134192107990830, 8.672969558257914, 9.013599181004549, 9.856421753472765, 10.49605814627023, 10.91060511901628, 11.51205887020577, 11.90570232390822, 12.36115708318212, 12.97485719960188, 13.49467581731726, 14.06331790052581, 14.73831272661168, 15.06129622412424

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