Properties

Label 2-7938-1.1-c1-0-92
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 4.37·5-s + 8-s + 4.37·10-s + 1.37·11-s − 2·13-s + 16-s + 1.37·17-s − 5·19-s + 4.37·20-s + 1.37·22-s + 1.62·23-s + 14.1·25-s − 2·26-s + 8.74·29-s − 2·31-s + 32-s + 1.37·34-s + 2·37-s − 5·38-s + 4.37·40-s + 4.62·41-s − 8.11·43-s + 1.37·44-s + 1.62·46-s + 14.1·50-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.95·5-s + 0.353·8-s + 1.38·10-s + 0.413·11-s − 0.554·13-s + 0.250·16-s + 0.332·17-s − 1.14·19-s + 0.977·20-s + 0.292·22-s + 0.339·23-s + 2.82·25-s − 0.392·26-s + 1.62·29-s − 0.359·31-s + 0.176·32-s + 0.235·34-s + 0.328·37-s − 0.811·38-s + 0.691·40-s + 0.722·41-s − 1.23·43-s + 0.206·44-s + 0.239·46-s + 1.99·50-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: \(0\)
Selberg data: \((2,\ 7938,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.289862871\)
\(L(\frac12)\) \(\approx\) \(5.289862871\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.37T + 5T^{2} \)
11 \( 1 - 1.37T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 1.37T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 - 1.62T + 23T^{2} \)
29 \( 1 - 8.74T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 4.62T + 41T^{2} \)
43 \( 1 + 8.11T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 8.74T + 53T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 + 3.11T + 61T^{2} \)
67 \( 1 + 2.11T + 67T^{2} \)
71 \( 1 - 7.11T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + 5.11T + 79T^{2} \)
83 \( 1 - 17.4T + 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 - 8.11T + 97T^{2} \)
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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

−7.67292806725829786819046708915, −6.80814352720580831078718649464, −6.28751289879770305519706067953, −5.88066197145615291718335526714, −4.95837757710970986866269041544, −4.62148075616987877061448022293, −3.40728342444072256634023222954, −2.56278118430626572680991745555, −2.00888100796704150334190993698, −1.08706750673603147258075008161, 1.08706750673603147258075008161, 2.00888100796704150334190993698, 2.56278118430626572680991745555, 3.40728342444072256634023222954, 4.62148075616987877061448022293, 4.95837757710970986866269041544, 5.88066197145615291718335526714, 6.28751289879770305519706067953, 6.80814352720580831078718649464, 7.67292806725829786819046708915

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