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Properties

Label 2-38115-1.1-c1-0-9
Degree $2$
Conductor $38115$
Sign $1$
Analytic cond. $304.349$
Root an. cond. $17.4456$
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-s − 4-s + 5-s − 7-s − 3·8-s + 10-s + 2·13-s − 14-s − 16-s + 6·17-s − 4·19-s − 20-s + 4·23-s + 25-s + 2·26-s + 28-s + 10·29-s − 4·31-s + 5·32-s + 6·34-s − 35-s − 2·37-s − 4·38-s − 3·40-s + 10·41-s − 12·43-s + 4·46-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.377·7-s − 1.06·8-s + 0.316·10-s + 0.554·13-s − 0.267·14-s − 1/4·16-s + 1.45·17-s − 0.917·19-s − 0.223·20-s + 0.834·23-s + 1/5·25-s + 0.392·26-s + 0.188·28-s + 1.85·29-s − 0.718·31-s + 0.883·32-s + 1.02·34-s − 0.169·35-s − 0.328·37-s − 0.648·38-s − 0.474·40-s + 1.56·41-s − 1.82·43-s + 0.589·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38115\)    =    \(3^{2} \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(304.349\)
Root analytic conductor: \(17.4456\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 38115,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.940469292\)
\(L(\frac12)\) \(\approx\) \(2.940469292\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 \)
good2 \( 1 - T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.75767525037512, −14.20885430910133, −13.88075913153497, −13.22896319558037, −12.83129137691636, −12.43335557468375, −11.92704085790548, −11.19891416035812, −10.62451128215824, −9.962209174162658, −9.661605242899338, −8.957305393167842, −8.446727706387882, −8.022248085386611, −7.072047599940292, −6.489764787487302, −6.049945865361782, −5.391471359077858, −4.957252561537985, −4.272765555159141, −3.561650095309919, −3.124469347498214, −2.419835324269949, −1.357081293099334, −0.6046894761928077, 0.6046894761928077, 1.357081293099334, 2.419835324269949, 3.124469347498214, 3.561650095309919, 4.272765555159141, 4.957252561537985, 5.391471359077858, 6.049945865361782, 6.489764787487302, 7.072047599940292, 8.022248085386611, 8.446727706387882, 8.957305393167842, 9.661605242899338, 9.962209174162658, 10.62451128215824, 11.19891416035812, 11.92704085790548, 12.43335557468375, 12.83129137691636, 13.22896319558037, 13.88075913153497, 14.20885430910133, 14.75767525037512

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