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Properties

Label 2-25392-1.1-c1-0-14
Degree $2$
Conductor $25392$
Sign $-1$
Analytic cond. $202.756$
Root an. cond. $14.2392$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s + 9-s − 4·11-s − 3·13-s − 8·17-s + 4·19-s + 21-s − 5·25-s − 27-s − 4·29-s + 8·31-s + 4·33-s − 37-s + 3·39-s + 8·41-s − 43-s + 6·47-s − 6·49-s + 8·51-s + 6·53-s − 4·57-s − 6·59-s + 15·61-s − 63-s + 13·67-s + 10·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.832·13-s − 1.94·17-s + 0.917·19-s + 0.218·21-s − 25-s − 0.192·27-s − 0.742·29-s + 1.43·31-s + 0.696·33-s − 0.164·37-s + 0.480·39-s + 1.24·41-s − 0.152·43-s + 0.875·47-s − 6/7·49-s + 1.12·51-s + 0.824·53-s − 0.529·57-s − 0.781·59-s + 1.92·61-s − 0.125·63-s + 1.58·67-s + 1.18·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25392\)    =    \(2^{4} \cdot 3 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(202.756\)
Root analytic conductor: \(14.2392\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 25392,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
23 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 15 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−15.68308734354372, −15.36910271700624, −14.57736774988742, −13.86721196321579, −13.35393576982197, −13.00973651719360, −12.40386097778348, −11.80690825953887, −11.24825480162840, −10.85484830766593, −10.06781308757976, −9.778236019481217, −9.138994989913521, −8.399335507392292, −7.743699581513858, −7.250764144697850, −6.643565126931835, −6.014812783463135, −5.378700478566508, −4.828100736071264, −4.246011098898579, −3.431888832125386, −2.430336910151028, −2.190476654120892, −0.7826482557901553, 0, 0.7826482557901553, 2.190476654120892, 2.430336910151028, 3.431888832125386, 4.246011098898579, 4.828100736071264, 5.378700478566508, 6.014812783463135, 6.643565126931835, 7.250764144697850, 7.743699581513858, 8.399335507392292, 9.138994989913521, 9.778236019481217, 10.06781308757976, 10.85484830766593, 11.24825480162840, 11.80690825953887, 12.40386097778348, 13.00973651719360, 13.35393576982197, 13.86721196321579, 14.57736774988742, 15.36910271700624, 15.68308734354372

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