Properties

Label 2-966-1.1-c3-0-63
Degree $2$
Conductor $966$
Sign $-1$
Analytic cond. $56.9958$
Root an. cond. $7.54955$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 4·4-s + 14.1·5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s + 28.2·10-s − 26.4·11-s − 12·12-s + 8.09·13-s − 14·14-s − 42.3·15-s + 16·16-s − 91.8·17-s + 18·18-s − 113.·19-s + 56.4·20-s + 21·21-s − 52.9·22-s − 23·23-s − 24·24-s + 74.4·25-s + 16.1·26-s − 27·27-s − 28·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.26·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.893·10-s − 0.725·11-s − 0.288·12-s + 0.172·13-s − 0.267·14-s − 0.729·15-s + 0.250·16-s − 1.30·17-s + 0.235·18-s − 1.36·19-s + 0.631·20-s + 0.218·21-s − 0.512·22-s − 0.208·23-s − 0.204·24-s + 0.595·25-s + 0.122·26-s − 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(56.9958\)
Root analytic conductor: \(7.54955\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 966,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 + 3T \)
7 \( 1 + 7T \)
23 \( 1 + 23T \)
good5 \( 1 - 14.1T + 125T^{2} \)
11 \( 1 + 26.4T + 1.33e3T^{2} \)
13 \( 1 - 8.09T + 2.19e3T^{2} \)
17 \( 1 + 91.8T + 4.91e3T^{2} \)
19 \( 1 + 113.T + 6.85e3T^{2} \)
29 \( 1 + 191.T + 2.43e4T^{2} \)
31 \( 1 - 47.5T + 2.97e4T^{2} \)
37 \( 1 + 80.9T + 5.06e4T^{2} \)
41 \( 1 - 141.T + 6.89e4T^{2} \)
43 \( 1 + 223.T + 7.95e4T^{2} \)
47 \( 1 - 33.0T + 1.03e5T^{2} \)
53 \( 1 - 76.0T + 1.48e5T^{2} \)
59 \( 1 + 548.T + 2.05e5T^{2} \)
61 \( 1 - 533.T + 2.26e5T^{2} \)
67 \( 1 - 625.T + 3.00e5T^{2} \)
71 \( 1 - 247.T + 3.57e5T^{2} \)
73 \( 1 + 956.T + 3.89e5T^{2} \)
79 \( 1 + 984.T + 4.93e5T^{2} \)
83 \( 1 - 188.T + 5.71e5T^{2} \)
89 \( 1 + 532.T + 7.04e5T^{2} \)
97 \( 1 + 115.T + 9.12e5T^{2} \)
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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

−9.391966982588854944744877282599, −8.433438565387808988206401206562, −7.14495219381952191876405284034, −6.35234490718405924939386712416, −5.81461799288397534398457153160, −4.95035628624719448009911918195, −3.98837135253394496452102690516, −2.55709907069374761343462109145, −1.78918646116632368496135172497, 0, 1.78918646116632368496135172497, 2.55709907069374761343462109145, 3.98837135253394496452102690516, 4.95035628624719448009911918195, 5.81461799288397534398457153160, 6.35234490718405924939386712416, 7.14495219381952191876405284034, 8.433438565387808988206401206562, 9.391966982588854944744877282599

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