Properties

Label 2-966-1.1-c3-0-15
Degree $2$
Conductor $966$
Sign $1$
Analytic cond. $56.9958$
Root an. cond. $7.54955$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 4·4-s − 9·5-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s − 18·10-s − 12·11-s − 12·12-s + 11·13-s + 14·14-s + 27·15-s + 16·16-s + 96·17-s + 18·18-s − 40·19-s − 36·20-s − 21·21-s − 24·22-s − 23·23-s − 24·24-s − 44·25-s + 22·26-s − 27·27-s + 28·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.804·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.569·10-s − 0.328·11-s − 0.288·12-s + 0.234·13-s + 0.267·14-s + 0.464·15-s + 1/4·16-s + 1.36·17-s + 0.235·18-s − 0.482·19-s − 0.402·20-s − 0.218·21-s − 0.232·22-s − 0.208·23-s − 0.204·24-s − 0.351·25-s + 0.165·26-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: yes
Arithmetic: yes
Character: $\chi_{966} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.279636721\)
\(L(\frac12)\) \(\approx\) \(2.279636721\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 + p T \)
7 \( 1 - p T \)
23 \( 1 + p T \)
good5 \( 1 + 9 T + p^{3} T^{2} \)
11 \( 1 + 12 T + p^{3} T^{2} \)
13 \( 1 - 11 T + p^{3} T^{2} \)
17 \( 1 - 96 T + p^{3} T^{2} \)
19 \( 1 + 40 T + p^{3} T^{2} \)
29 \( 1 + 231 T + p^{3} T^{2} \)
31 \( 1 + 94 T + p^{3} T^{2} \)
37 \( 1 - 47 T + p^{3} T^{2} \)
41 \( 1 + 27 T + p^{3} T^{2} \)
43 \( 1 - 479 T + p^{3} T^{2} \)
47 \( 1 - 9 p T + p^{3} T^{2} \)
53 \( 1 - 516 T + p^{3} T^{2} \)
59 \( 1 - 882 T + p^{3} T^{2} \)
61 \( 1 - 842 T + p^{3} T^{2} \)
67 \( 1 + 844 T + p^{3} T^{2} \)
71 \( 1 - 654 T + p^{3} T^{2} \)
73 \( 1 + 496 T + p^{3} T^{2} \)
79 \( 1 - 260 T + p^{3} T^{2} \)
83 \( 1 + 156 T + p^{3} T^{2} \)
89 \( 1 + 414 T + p^{3} T^{2} \)
97 \( 1 - 1343 T + p^{3} T^{2} \)
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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

−9.875172665982467739353431091583, −8.648406082251654669621302906439, −7.64343984002500161580691067550, −7.22125505693524235797765894725, −5.85901961075733152316593624022, −5.41845604601719456519099497286, −4.21102459985340616548126591765, −3.62878726202766341279771401105, −2.18702741690359664785202207519, −0.75259941511163770091287571706, 0.75259941511163770091287571706, 2.18702741690359664785202207519, 3.62878726202766341279771401105, 4.21102459985340616548126591765, 5.41845604601719456519099497286, 5.85901961075733152316593624022, 7.22125505693524235797765894725, 7.64343984002500161580691067550, 8.648406082251654669621302906439, 9.875172665982467739353431091583

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