Properties

Label 2-945-1.1-c1-0-11
Degree $2$
Conductor $945$
Sign $1$
Analytic cond. $7.54586$
Root an. cond. $2.74697$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.857·2-s − 1.26·4-s − 5-s + 7-s − 2.79·8-s − 0.857·10-s − 1.40·11-s + 5.12·13-s + 0.857·14-s + 0.128·16-s + 5.92·17-s − 2.51·19-s + 1.26·20-s − 1.20·22-s + 5.92·23-s + 25-s + 4.39·26-s − 1.26·28-s − 4.52·29-s + 9.63·31-s + 5.70·32-s + 5.07·34-s − 35-s + 0.284·37-s − 2.15·38-s + 2.79·40-s + 1.60·41-s + ⋯
L(s)  = 1  + 0.606·2-s − 0.632·4-s − 0.447·5-s + 0.377·7-s − 0.989·8-s − 0.271·10-s − 0.424·11-s + 1.42·13-s + 0.229·14-s + 0.0320·16-s + 1.43·17-s − 0.576·19-s + 0.282·20-s − 0.257·22-s + 1.23·23-s + 0.200·25-s + 0.861·26-s − 0.238·28-s − 0.841·29-s + 1.73·31-s + 1.00·32-s + 0.870·34-s − 0.169·35-s + 0.0468·37-s − 0.349·38-s + 0.442·40-s + 0.251·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.747264173\)
\(L(\frac12)\) \(\approx\) \(1.747264173\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
7 \( 1 - T \)
good2 \( 1 - 0.857T + 2T^{2} \)
11 \( 1 + 1.40T + 11T^{2} \)
13 \( 1 - 5.12T + 13T^{2} \)
17 \( 1 - 5.92T + 17T^{2} \)
19 \( 1 + 2.51T + 19T^{2} \)
23 \( 1 - 5.92T + 23T^{2} \)
29 \( 1 + 4.52T + 29T^{2} \)
31 \( 1 - 9.63T + 31T^{2} \)
37 \( 1 - 0.284T + 37T^{2} \)
41 \( 1 - 1.60T + 41T^{2} \)
43 \( 1 - 12.6T + 43T^{2} \)
47 \( 1 + 7.82T + 47T^{2} \)
53 \( 1 + 7.32T + 53T^{2} \)
59 \( 1 + 1.71T + 59T^{2} \)
61 \( 1 + 3.92T + 61T^{2} \)
67 \( 1 - 15.4T + 67T^{2} \)
71 \( 1 - 11.6T + 71T^{2} \)
73 \( 1 + 7.12T + 73T^{2} \)
79 \( 1 + 7.35T + 79T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 + 3.53T + 89T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−10.06892758498169464186270162535, −9.100772736050595169844102114780, −8.344712968266660036718619267069, −7.70190171440030512149615838429, −6.38673286049943538830405322446, −5.57586001860482077141586743162, −4.70527548286844812828528285987, −3.80888077538634852637200363057, −2.95874398382912614385956150552, −1.01672450645728970984382158112, 1.01672450645728970984382158112, 2.95874398382912614385956150552, 3.80888077538634852637200363057, 4.70527548286844812828528285987, 5.57586001860482077141586743162, 6.38673286049943538830405322446, 7.70190171440030512149615838429, 8.344712968266660036718619267069, 9.100772736050595169844102114780, 10.06892758498169464186270162535

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