Properties

Label 2-7942-1.1-c1-0-139
Degree $2$
Conductor $7942$
Sign $-1$
Analytic cond. $63.4171$
Root an. cond. $7.96349$
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 − 2.89·3-s + 4-s + 2.32·5-s + 2.89·6-s + 0.619·7-s − 8-s + 5.36·9-s − 2.32·10-s − 11-s − 2.89·12-s − 5.40·13-s − 0.619·14-s − 6.71·15-s + 16-s − 2.11·17-s − 5.36·18-s + 2.32·20-s − 1.79·21-s + 22-s + 1.91·23-s + 2.89·24-s + 0.389·25-s + 5.40·26-s − 6.83·27-s + 0.619·28-s + 8.17·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.66·3-s + 0.5·4-s + 1.03·5-s + 1.18·6-s + 0.234·7-s − 0.353·8-s + 1.78·9-s − 0.734·10-s − 0.301·11-s − 0.834·12-s − 1.49·13-s − 0.165·14-s − 1.73·15-s + 0.250·16-s − 0.511·17-s − 1.26·18-s + 0.519·20-s − 0.391·21-s + 0.213·22-s + 0.399·23-s + 0.590·24-s + 0.0779·25-s + 1.06·26-s − 1.31·27-s + 0.117·28-s + 1.51·29-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: no
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 \( 1 + 2.89T + 3T^{2} \)
5 \( 1 - 2.32T + 5T^{2} \)
7 \( 1 - 0.619T + 7T^{2} \)
13 \( 1 + 5.40T + 13T^{2} \)
17 \( 1 + 2.11T + 17T^{2} \)
23 \( 1 - 1.91T + 23T^{2} \)
29 \( 1 - 8.17T + 29T^{2} \)
31 \( 1 + 2.15T + 31T^{2} \)
37 \( 1 + 8.95T + 37T^{2} \)
41 \( 1 - 3.78T + 41T^{2} \)
43 \( 1 - 4.48T + 43T^{2} \)
47 \( 1 - 6.91T + 47T^{2} \)
53 \( 1 - 13.0T + 53T^{2} \)
59 \( 1 + 0.481T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 - 0.861T + 71T^{2} \)
73 \( 1 - 0.742T + 73T^{2} \)
79 \( 1 - 13.7T + 79T^{2} \)
83 \( 1 + 9.63T + 83T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 + 8.77T + 97T^{2} \)
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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

−7.20833129687096225068137369458, −6.83327642766740145425205049090, −6.13323687721497451493481986605, −5.40577887945898165078048880435, −5.06001561399428859446247039435, −4.22027969823775315664861306045, −2.71500392316906545885466643781, −2.00829884091825367709147625389, −1.00928750771960211338301986773, 0, 1.00928750771960211338301986773, 2.00829884091825367709147625389, 2.71500392316906545885466643781, 4.22027969823775315664861306045, 5.06001561399428859446247039435, 5.40577887945898165078048880435, 6.13323687721497451493481986605, 6.83327642766740145425205049090, 7.20833129687096225068137369458

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