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Properties

Label 2-7942-1.1-c1-0-153
Degree $2$
Conductor $7942$
Sign $-1$
Analytic cond. $63.4171$
Root an. cond. $7.96349$
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 + 4-s − 4·5-s − 4·7-s + 8-s − 3·9-s − 4·10-s + 11-s + 7·13-s − 4·14-s + 16-s − 3·18-s − 4·20-s + 22-s + 4·23-s + 11·25-s + 7·26-s − 4·28-s − 3·29-s + 32-s + 16·35-s − 3·36-s + 8·37-s − 4·40-s − 12·41-s + 7·43-s + 44-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.78·5-s − 1.51·7-s + 0.353·8-s − 9-s − 1.26·10-s + 0.301·11-s + 1.94·13-s − 1.06·14-s + 1/4·16-s − 0.707·18-s − 0.894·20-s + 0.213·22-s + 0.834·23-s + 11/5·25-s + 1.37·26-s − 0.755·28-s − 0.557·29-s + 0.176·32-s + 2.70·35-s − 1/2·36-s + 1.31·37-s − 0.632·40-s − 1.87·41-s + 1.06·43-s + 0.150·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7942\)    =    \(2 \cdot 11 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(63.4171\)
Root analytic conductor: \(7.96349\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7942,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 - T \)
19 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−7.35921014049096831547863072082, −6.69033182920975630710534486503, −6.17257156415181068988122114210, −5.48272456049391023053489696435, −4.40638984445877629161596984965, −3.71838898702585309982223219300, −3.36922710853353974840771616403, −2.77397918316604924788381349532, −1.07184012577794186254168194107, 0, 1.07184012577794186254168194107, 2.77397918316604924788381349532, 3.36922710853353974840771616403, 3.71838898702585309982223219300, 4.40638984445877629161596984965, 5.48272456049391023053489696435, 6.17257156415181068988122114210, 6.69033182920975630710534486503, 7.35921014049096831547863072082

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