Properties

Label 2-38962-1.1-c1-0-26
Degree $2$
Conductor $38962$
Sign $-1$
Analytic cond. $311.113$
Root an. cond. $17.6383$
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  + 2-s + 2·3-s + 4-s + 2·6-s − 7-s + 8-s + 9-s + 2·12-s − 14-s + 16-s − 6·17-s + 18-s + 6·19-s − 2·21-s − 23-s + 2·24-s − 5·25-s − 4·27-s − 28-s − 10·29-s + 4·31-s + 32-s − 6·34-s + 36-s − 2·37-s + 6·38-s + 10·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.577·12-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 1.37·19-s − 0.436·21-s − 0.208·23-s + 0.408·24-s − 25-s − 0.769·27-s − 0.188·28-s − 1.85·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s + 0.973·38-s + 1.56·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38962 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38962 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38962\)    =    \(2 \cdot 7 \cdot 11^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(311.113\)
Root analytic conductor: \(17.6383\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{38962} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 38962,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 \)
23 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−15.10435161514839, −14.42575324001251, −13.93840533652546, −13.60067729393900, −13.26290732531130, −12.58219268539389, −12.10842423596321, −11.33410231928611, −11.10782612359879, −10.33317302026786, −9.528071074064114, −9.297228395973814, −8.806092427250881, −7.925384522256461, −7.594343821258102, −7.111054752837606, −6.261766239680081, −5.804668820112033, −5.181738923659043, −4.291303380076271, −3.857988511766195, −3.333028082805962, −2.471253044545042, −2.283915748960876, −1.271193741375822, 0, 1.271193741375822, 2.283915748960876, 2.471253044545042, 3.333028082805962, 3.857988511766195, 4.291303380076271, 5.181738923659043, 5.804668820112033, 6.261766239680081, 7.111054752837606, 7.594343821258102, 7.925384522256461, 8.806092427250881, 9.297228395973814, 9.528071074064114, 10.33317302026786, 11.10782612359879, 11.33410231928611, 12.10842423596321, 12.58219268539389, 13.26290732531130, 13.60067729393900, 13.93840533652546, 14.42575324001251, 15.10435161514839

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