Properties

Label 2-966-1.1-c5-0-106
Degree $2$
Conductor $966$
Sign $-1$
Analytic cond. $154.930$
Root an. cond. $12.4471$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 9·3-s + 16·4-s + 28·5-s + 36·6-s + 49·7-s + 64·8-s + 81·9-s + 112·10-s − 15·11-s + 144·12-s − 1.06e3·13-s + 196·14-s + 252·15-s + 256·16-s − 80·17-s + 324·18-s − 1.31e3·19-s + 448·20-s + 441·21-s − 60·22-s + 529·23-s + 576·24-s − 2.34e3·25-s − 4.26e3·26-s + 729·27-s + 784·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.500·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.354·10-s − 0.0373·11-s + 0.288·12-s − 1.74·13-s + 0.267·14-s + 0.289·15-s + 1/4·16-s − 0.0671·17-s + 0.235·18-s − 0.838·19-s + 0.250·20-s + 0.218·21-s − 0.0264·22-s + 0.208·23-s + 0.204·24-s − 0.749·25-s − 1.23·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(154.930\)
Root analytic conductor: \(12.4471\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: $\chi_{966} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 966,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{2} T \)
3 \( 1 - p^{2} T \)
7 \( 1 - p^{2} T \)
23 \( 1 - p^{2} T \)
good5 \( 1 - 28 T + p^{5} T^{2} \)
11 \( 1 + 15 T + p^{5} T^{2} \)
13 \( 1 + 82 p T + p^{5} T^{2} \)
17 \( 1 + 80 T + p^{5} T^{2} \)
19 \( 1 + 1319 T + p^{5} T^{2} \)
29 \( 1 + 3586 T + p^{5} T^{2} \)
31 \( 1 + 7850 T + p^{5} T^{2} \)
37 \( 1 - 5074 T + p^{5} T^{2} \)
41 \( 1 + 17627 T + p^{5} T^{2} \)
43 \( 1 - 10392 T + p^{5} T^{2} \)
47 \( 1 + 11805 T + p^{5} T^{2} \)
53 \( 1 + 345 T + p^{5} T^{2} \)
59 \( 1 + 31347 T + p^{5} T^{2} \)
61 \( 1 + 5019 T + p^{5} T^{2} \)
67 \( 1 + 26552 T + p^{5} T^{2} \)
71 \( 1 + 23492 T + p^{5} T^{2} \)
73 \( 1 + 5356 T + p^{5} T^{2} \)
79 \( 1 - 99870 T + p^{5} T^{2} \)
83 \( 1 - 47482 T + p^{5} T^{2} \)
89 \( 1 + 58182 T + p^{5} T^{2} \)
97 \( 1 - 72722 T + p^{5} T^{2} \)
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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

−8.954889792762959753404528229900, −7.79663162520243274166295578797, −7.25926427979336317174411803352, −6.20996055136163310443195120952, −5.22744730135195268086512445649, −4.51360503254101463857959842309, −3.44600011807659256509426801215, −2.33375619470170670960788654688, −1.77037186081357051342176995070, 0, 1.77037186081357051342176995070, 2.33375619470170670960788654688, 3.44600011807659256509426801215, 4.51360503254101463857959842309, 5.22744730135195268086512445649, 6.20996055136163310443195120952, 7.25926427979336317174411803352, 7.79663162520243274166295578797, 8.954889792762959753404528229900

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