Properties

Label 2-931-1.1-c1-0-48
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.37·2-s + 0.579·3-s + 3.66·4-s + 3.67·5-s + 1.37·6-s + 3.95·8-s − 2.66·9-s + 8.74·10-s − 4.50·11-s + 2.12·12-s + 6.04·13-s + 2.13·15-s + 2.09·16-s − 5.29·17-s − 6.33·18-s − 19-s + 13.4·20-s − 10.7·22-s − 0.432·23-s + 2.29·24-s + 8.51·25-s + 14.3·26-s − 3.28·27-s + 1.82·29-s + 5.07·30-s − 5.46·31-s − 2.93·32-s + ⋯
L(s)  = 1  + 1.68·2-s + 0.334·3-s + 1.83·4-s + 1.64·5-s + 0.563·6-s + 1.39·8-s − 0.887·9-s + 2.76·10-s − 1.35·11-s + 0.613·12-s + 1.67·13-s + 0.550·15-s + 0.523·16-s − 1.28·17-s − 1.49·18-s − 0.229·19-s + 3.01·20-s − 2.28·22-s − 0.0901·23-s + 0.468·24-s + 1.70·25-s + 2.82·26-s − 0.631·27-s + 0.339·29-s + 0.926·30-s − 0.981·31-s − 0.518·32-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

\(L(1)\) \(\approx\) \(5.209305603\)
\(L(\frac12)\) \(\approx\) \(5.209305603\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.37T + 2T^{2} \)
3 \( 1 - 0.579T + 3T^{2} \)
5 \( 1 - 3.67T + 5T^{2} \)
11 \( 1 + 4.50T + 11T^{2} \)
13 \( 1 - 6.04T + 13T^{2} \)
17 \( 1 + 5.29T + 17T^{2} \)
23 \( 1 + 0.432T + 23T^{2} \)
29 \( 1 - 1.82T + 29T^{2} \)
31 \( 1 + 5.46T + 31T^{2} \)
37 \( 1 + 6.12T + 37T^{2} \)
41 \( 1 + 1.21T + 41T^{2} \)
43 \( 1 - 2.24T + 43T^{2} \)
47 \( 1 - 9.90T + 47T^{2} \)
53 \( 1 - 10.2T + 53T^{2} \)
59 \( 1 - 1.30T + 59T^{2} \)
61 \( 1 - 7.92T + 61T^{2} \)
67 \( 1 + 9.41T + 67T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 + 3.45T + 73T^{2} \)
79 \( 1 + 0.0567T + 79T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 + 1.74T + 97T^{2} \)
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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

−10.55798277575757866435349508927, −9.078522215198418475174979040742, −8.556483894165907054481720415051, −7.10316906777054480640179229424, −6.07156263950605370390598995527, −5.76372747562437577054107730912, −4.95694348698095661747019535300, −3.71251907039641648766382173956, −2.66262312626001746458664970282, −2.02571343941637063035287173026, 2.02571343941637063035287173026, 2.66262312626001746458664970282, 3.71251907039641648766382173956, 4.95694348698095661747019535300, 5.76372747562437577054107730912, 6.07156263950605370390598995527, 7.10316906777054480640179229424, 8.556483894165907054481720415051, 9.078522215198418475174979040742, 10.55798277575757866435349508927

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