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Properties

Label 2-2151-239.238-c0-0-2
Degree $2$
Conductor $2151$
Sign $1$
Analytic cond. $1.07348$
Root an. cond. $1.03609$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.618·2-s − 0.618·4-s − 0.618·5-s + 8-s + 0.381·10-s + 1.61·11-s − 2·17-s + 0.381·20-s − 1.00·22-s − 0.618·25-s + 1.61·29-s + 0.618·31-s − 0.999·32-s + 1.23·34-s − 0.618·40-s − 0.999·44-s + 49-s + 0.381·50-s − 1.00·55-s − 1.00·58-s + 2·61-s − 0.381·62-s + 0.618·64-s + 2·67-s + 1.23·68-s + 1.61·71-s + 1.61·83-s + ⋯
L(s)  = 1  − 0.618·2-s − 0.618·4-s − 0.618·5-s + 8-s + 0.381·10-s + 1.61·11-s − 2·17-s + 0.381·20-s − 1.00·22-s − 0.618·25-s + 1.61·29-s + 0.618·31-s − 0.999·32-s + 1.23·34-s − 0.618·40-s − 0.999·44-s + 49-s + 0.381·50-s − 1.00·55-s − 1.00·58-s + 2·61-s − 0.381·62-s + 0.618·64-s + 2·67-s + 1.23·68-s + 1.61·71-s + 1.61·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $1$
Analytic conductor: \(1.07348\)
Root analytic conductor: \(1.03609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (955, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6561376524\)
\(L(\frac12)\) \(\approx\) \(0.6561376524\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
239 \( 1 + T \)
good2 \( 1 + 0.618T + T^{2} \)
5 \( 1 + 0.618T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 1.61T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 2T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 1.61T + T^{2} \)
31 \( 1 - 0.618T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 2T + T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 - 1.61T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.61T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.171472081492781686708045463320, −8.550239327567276490040822434967, −8.023186007479946412827564376962, −6.86465073720404434702563489460, −6.49758097879193600084400970858, −5.10119724032208083074151577129, −4.21566624381821295087942567330, −3.83251471585125388032081975924, −2.24237372544536865240578055503, −0.893201013245216599471994312764, 0.893201013245216599471994312764, 2.24237372544536865240578055503, 3.83251471585125388032081975924, 4.21566624381821295087942567330, 5.10119724032208083074151577129, 6.49758097879193600084400970858, 6.86465073720404434702563489460, 8.023186007479946412827564376962, 8.550239327567276490040822434967, 9.171472081492781686708045463320

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