Properties

Label 2-7942-1.1-c1-0-270
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 + 2·3-s + 4-s + 2·5-s − 2·6-s + 2·7-s − 8-s + 9-s − 2·10-s − 11-s + 2·12-s − 2·13-s − 2·14-s + 4·15-s + 16-s − 6·17-s − 18-s + 2·20-s + 4·21-s + 22-s − 2·24-s − 25-s + 2·26-s − 4·27-s + 2·28-s − 2·29-s − 4·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s − 0.816·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s + 0.577·12-s − 0.554·13-s − 0.534·14-s + 1.03·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.447·20-s + 0.872·21-s + 0.213·22-s − 0.408·24-s − 1/5·25-s + 0.392·26-s − 0.769·27-s + 0.377·28-s − 0.371·29-s − 0.730·30-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 \( 1 - 2 T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−7.69938598332186259867216934677, −7.02553326906149803508105968685, −6.27069307967945204937986403584, −5.40949627748415142883247623962, −4.72024103200300233436214624959, −3.68951187254623807673954852442, −2.79745035634974018820852356212, −2.02878714193227454534285498009, −1.72536263148644519711541882318, 0, 1.72536263148644519711541882318, 2.02878714193227454534285498009, 2.79745035634974018820852356212, 3.68951187254623807673954852442, 4.72024103200300233436214624959, 5.40949627748415142883247623962, 6.27069307967945204937986403584, 7.02553326906149803508105968685, 7.69938598332186259867216934677

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