Properties

Label 2-38808-1.1-c1-0-13
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  − 2·5-s + 11-s + 2·13-s + 2·17-s − 4·19-s + 6·23-s − 25-s + 6·29-s − 10·31-s + 2·37-s − 2·41-s + 10·43-s + 6·53-s − 2·55-s − 12·59-s − 2·61-s − 4·65-s − 12·67-s − 2·71-s − 2·73-s − 14·83-s − 4·85-s + 10·89-s + 8·95-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.301·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s + 1.25·23-s − 1/5·25-s + 1.11·29-s − 1.79·31-s + 0.328·37-s − 0.312·41-s + 1.52·43-s + 0.824·53-s − 0.269·55-s − 1.56·59-s − 0.256·61-s − 0.496·65-s − 1.46·67-s − 0.237·71-s − 0.234·73-s − 1.53·83-s − 0.433·85-s + 1.05·89-s + 0.820·95-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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

\(L(1)\) \(\approx\) \(1.669397893\)
\(L(\frac12)\) \(\approx\) \(1.669397893\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−14.86401908362311, −14.37664005299347, −13.84991119538777, −13.13965894861698, −12.72985139004330, −12.20426213831176, −11.70353332145116, −11.07398144330789, −10.78527112727169, −10.17692858265780, −9.416987808471111, −8.729747109380197, −8.656016774828360, −7.607547030412235, −7.482789953397855, −6.746526593031282, −6.035450690865006, −5.614464619163172, −4.601268156102320, −4.357384654892732, −3.507685168977923, −3.132324281166917, −2.175349944275337, −1.328185863889487, −0.5049791445860975, 0.5049791445860975, 1.328185863889487, 2.175349944275337, 3.132324281166917, 3.507685168977923, 4.357384654892732, 4.601268156102320, 5.614464619163172, 6.035450690865006, 6.746526593031282, 7.482789953397855, 7.607547030412235, 8.656016774828360, 8.729747109380197, 9.416987808471111, 10.17692858265780, 10.78527112727169, 11.07398144330789, 11.70353332145116, 12.20426213831176, 12.72985139004330, 13.13965894861698, 13.84991119538777, 14.37664005299347, 14.86401908362311

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