Properties

Label 2-38291-1.1-c1-0-0
Degree $2$
Conductor $38291$
Sign $1$
Analytic cond. $305.755$
Root an. cond. $17.4858$
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  + 2·2-s − 3-s + 2·4-s + 5-s − 2·6-s − 2·7-s − 2·9-s + 2·10-s − 11-s − 2·12-s − 4·13-s − 4·14-s − 15-s − 4·16-s − 2·17-s − 4·18-s + 2·20-s + 2·21-s − 2·22-s + 23-s − 4·25-s − 8·26-s + 5·27-s − 4·28-s − 2·30-s − 7·31-s − 8·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s − 0.755·7-s − 2/3·9-s + 0.632·10-s − 0.301·11-s − 0.577·12-s − 1.10·13-s − 1.06·14-s − 0.258·15-s − 16-s − 0.485·17-s − 0.942·18-s + 0.447·20-s + 0.436·21-s − 0.426·22-s + 0.208·23-s − 4/5·25-s − 1.56·26-s + 0.962·27-s − 0.755·28-s − 0.365·30-s − 1.25·31-s − 1.41·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38291 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38291 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38291\)    =    \(11 \cdot 59^{2}\)
Sign: $1$
Analytic conductor: \(305.755\)
Root analytic conductor: \(17.4858\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 38291,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3798434933\)
\(L(\frac12)\) \(\approx\) \(0.3798434933\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T \)
59 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 + T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
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

−14.71217477830291, −14.24775689029730, −13.77355401309867, −13.27019990544135, −12.63509788288163, −12.55067048496360, −11.74438948441375, −11.44552864958581, −10.80148897841731, −10.19855754357907, −9.552668906443518, −9.142707531607480, −8.430830990454663, −7.601376385687528, −6.905870736094264, −6.521645931517473, −5.761081129215114, −5.637918074886440, −4.828482711056518, −4.539379309149662, −3.489538831438976, −3.144377853174592, −2.416749976871860, −1.768860897655281, −0.1679845147825816, 0.1679845147825816, 1.768860897655281, 2.416749976871860, 3.144377853174592, 3.489538831438976, 4.539379309149662, 4.828482711056518, 5.637918074886440, 5.761081129215114, 6.521645931517473, 6.905870736094264, 7.601376385687528, 8.430830990454663, 9.142707531607480, 9.552668906443518, 10.19855754357907, 10.80148897841731, 11.44552864958581, 11.74438948441375, 12.55067048496360, 12.63509788288163, 13.27019990544135, 13.77355401309867, 14.24775689029730, 14.71217477830291

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