Properties

Label 2-931-1.1-c1-0-26
Degree $2$
Conductor $931$
Sign $1$
Analytic cond. $7.43407$
Root an. cond. $2.72654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.20·2-s + 2.62·3-s + 2.84·4-s + 4.23·5-s − 5.77·6-s − 1.85·8-s + 3.89·9-s − 9.32·10-s − 4.54·11-s + 7.46·12-s + 0.667·13-s + 11.1·15-s − 1.59·16-s − 1.86·17-s − 8.57·18-s + 19-s + 12.0·20-s + 10.0·22-s + 9.29·23-s − 4.88·24-s + 12.9·25-s − 1.46·26-s + 2.34·27-s − 1.31·29-s − 24.4·30-s − 3.90·31-s + 7.23·32-s + ⋯
L(s)  = 1  − 1.55·2-s + 1.51·3-s + 1.42·4-s + 1.89·5-s − 2.35·6-s − 0.657·8-s + 1.29·9-s − 2.94·10-s − 1.37·11-s + 2.15·12-s + 0.185·13-s + 2.87·15-s − 0.398·16-s − 0.452·17-s − 2.02·18-s + 0.229·19-s + 2.69·20-s + 2.13·22-s + 1.93·23-s − 0.996·24-s + 2.59·25-s − 0.287·26-s + 0.451·27-s − 0.243·29-s − 4.47·30-s − 0.701·31-s + 1.27·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.685839484\)
\(L(\frac12)\) \(\approx\) \(1.685839484\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 - T \)
good2 \( 1 + 2.20T + 2T^{2} \)
3 \( 1 - 2.62T + 3T^{2} \)
5 \( 1 - 4.23T + 5T^{2} \)
11 \( 1 + 4.54T + 11T^{2} \)
13 \( 1 - 0.667T + 13T^{2} \)
17 \( 1 + 1.86T + 17T^{2} \)
23 \( 1 - 9.29T + 23T^{2} \)
29 \( 1 + 1.31T + 29T^{2} \)
31 \( 1 + 3.90T + 31T^{2} \)
37 \( 1 - 3.00T + 37T^{2} \)
41 \( 1 - 5.48T + 41T^{2} \)
43 \( 1 - 4.44T + 43T^{2} \)
47 \( 1 - 5.81T + 47T^{2} \)
53 \( 1 + 9.95T + 53T^{2} \)
59 \( 1 + 2.07T + 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 + 0.523T + 67T^{2} \)
71 \( 1 + 4.03T + 71T^{2} \)
73 \( 1 + 5.38T + 73T^{2} \)
79 \( 1 + 2.64T + 79T^{2} \)
83 \( 1 + 2.83T + 83T^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 - 3.71T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.661469803557087100254476674089, −9.171017391679634324893167108471, −8.790972861900484948219028495394, −7.76843235287944335414981617370, −7.15029632277623021736208618344, −6.00754510708544070396967427415, −4.87310427508501378049725696717, −2.89646740139894406007331817008, −2.37665577815430665377409147267, −1.38236016424032883806659634406, 1.38236016424032883806659634406, 2.37665577815430665377409147267, 2.89646740139894406007331817008, 4.87310427508501378049725696717, 6.00754510708544070396967427415, 7.15029632277623021736208618344, 7.76843235287944335414981617370, 8.790972861900484948219028495394, 9.171017391679634324893167108471, 9.661469803557087100254476674089

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