Properties

Label 2-38808-1.1-c1-0-17
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 11-s + 2·13-s + 2·17-s + 4·19-s − 5·25-s + 6·29-s + 2·31-s − 2·37-s − 6·41-s − 6·47-s − 2·53-s + 2·59-s − 2·61-s − 4·67-s − 6·73-s + 12·79-s − 4·83-s + 8·89-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.301·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s − 25-s + 1.11·29-s + 0.359·31-s − 0.328·37-s − 0.937·41-s − 0.875·47-s − 0.274·53-s + 0.260·59-s − 0.256·61-s − 0.488·67-s − 0.702·73-s + 1.35·79-s − 0.439·83-s + 0.847·89-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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: \(0\)
Selberg data: \((2,\ 38808,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.514931468\)
\(L(\frac12)\) \(\approx\) \(2.514931468\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−14.84668301352871, −14.13726819759417, −13.86765379130687, −13.35843977316015, −12.74172011900801, −12.11095980017554, −11.66011004253973, −11.35754956484492, −10.43029916155578, −10.15116663375399, −9.562804115156188, −8.940475780228495, −8.425880381143168, −7.809765190478768, −7.369586042285162, −6.538503380346363, −6.205014802509760, −5.465206746977730, −4.904989984492494, −4.247454177912366, −3.433971540628066, −3.127003965931572, −2.099407843399749, −1.404023873663841, −0.6121399461500174, 0.6121399461500174, 1.404023873663841, 2.099407843399749, 3.127003965931572, 3.433971540628066, 4.247454177912366, 4.904989984492494, 5.465206746977730, 6.205014802509760, 6.538503380346363, 7.369586042285162, 7.809765190478768, 8.425880381143168, 8.940475780228495, 9.562804115156188, 10.15116663375399, 10.43029916155578, 11.35754956484492, 11.66011004253973, 12.11095980017554, 12.74172011900801, 13.35843977316015, 13.86765379130687, 14.13726819759417, 14.84668301352871

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