Properties

Label 2-7938-1.1-c1-0-81
Degree $2$
Conductor $7938$
Sign $-1$
Analytic cond. $63.3852$
Root an. cond. $7.96148$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 1.44·5-s − 8-s + 1.44·10-s − 2·11-s − 4.89·13-s + 16-s + 2·17-s − 2.55·19-s − 1.44·20-s + 2·22-s + 23-s − 2.89·25-s + 4.89·26-s + 6.89·29-s − 6·31-s − 32-s − 2·34-s + 11.7·37-s + 2.55·38-s + 1.44·40-s + 9.79·41-s + 6.89·43-s − 2·44-s − 46-s + 9.79·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.648·5-s − 0.353·8-s + 0.458·10-s − 0.603·11-s − 1.35·13-s + 0.250·16-s + 0.485·17-s − 0.585·19-s − 0.324·20-s + 0.426·22-s + 0.208·23-s − 0.579·25-s + 0.960·26-s + 1.28·29-s − 1.07·31-s − 0.176·32-s − 0.342·34-s + 1.93·37-s + 0.413·38-s + 0.229·40-s + 1.53·41-s + 1.05·43-s − 0.301·44-s − 0.147·46-s + 1.42·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7938\)    =    \(2 \cdot 3^{4} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(63.3852\)
Root analytic conductor: \(7.96148\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7938,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 1.44T + 5T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 4.89T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 2.55T + 19T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 - 6.89T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 - 11.7T + 37T^{2} \)
41 \( 1 - 9.79T + 41T^{2} \)
43 \( 1 - 6.89T + 43T^{2} \)
47 \( 1 - 9.79T + 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 + 2T + 59T^{2} \)
61 \( 1 + 6.55T + 61T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 + 0.101T + 71T^{2} \)
73 \( 1 - 6.89T + 73T^{2} \)
79 \( 1 + 1.89T + 79T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 + 2.89T + 97T^{2} \)
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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

−7.51520901043969511407660084003, −7.21533349882079743176154591595, −6.12406417185450335289190515018, −5.54802319154076693826343105147, −4.55550698621807828327975736456, −3.99145639330950275952539782242, −2.74983060598549025951020948510, −2.40150637121654632253891681281, −1.00607691179480837755584785816, 0, 1.00607691179480837755584785816, 2.40150637121654632253891681281, 2.74983060598549025951020948510, 3.99145639330950275952539782242, 4.55550698621807828327975736456, 5.54802319154076693826343105147, 6.12406417185450335289190515018, 7.21533349882079743176154591595, 7.51520901043969511407660084003

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