Properties

Label 2-931-1.1-c1-0-47
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  + 1.30·2-s + 3.30·3-s − 0.302·4-s + 3·5-s + 4.30·6-s − 3·8-s + 7.90·9-s + 3.90·10-s − 4.30·11-s − 1.00·12-s − 1.60·13-s + 9.90·15-s − 3.30·16-s + 1.69·17-s + 10.3·18-s − 19-s − 0.908·20-s − 5.60·22-s − 3·23-s − 9.90·24-s + 4·25-s − 2.09·26-s + 16.2·27-s − 0.908·29-s + 12.9·30-s + 2.30·31-s + 1.69·32-s + ⋯
L(s)  = 1  + 0.921·2-s + 1.90·3-s − 0.151·4-s + 1.34·5-s + 1.75·6-s − 1.06·8-s + 2.63·9-s + 1.23·10-s − 1.29·11-s − 0.288·12-s − 0.445·13-s + 2.55·15-s − 0.825·16-s + 0.411·17-s + 2.42·18-s − 0.229·19-s − 0.203·20-s − 1.19·22-s − 0.625·23-s − 2.02·24-s + 0.800·25-s − 0.410·26-s + 3.11·27-s − 0.168·29-s + 2.35·30-s + 0.413·31-s + 0.300·32-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

\(L(1)\) \(\approx\) \(4.643618647\)
\(L(\frac12)\) \(\approx\) \(4.643618647\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.30T + 2T^{2} \)
3 \( 1 - 3.30T + 3T^{2} \)
5 \( 1 - 3T + 5T^{2} \)
11 \( 1 + 4.30T + 11T^{2} \)
13 \( 1 + 1.60T + 13T^{2} \)
17 \( 1 - 1.69T + 17T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 + 0.908T + 29T^{2} \)
31 \( 1 - 2.30T + 31T^{2} \)
37 \( 1 + 3.60T + 37T^{2} \)
41 \( 1 + 4.30T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 - 8.21T + 47T^{2} \)
53 \( 1 - 3.90T + 53T^{2} \)
59 \( 1 + 8.21T + 59T^{2} \)
61 \( 1 + 10.2T + 61T^{2} \)
67 \( 1 - 8.90T + 67T^{2} \)
71 \( 1 + 2.21T + 71T^{2} \)
73 \( 1 - 16.5T + 73T^{2} \)
79 \( 1 + 3.21T + 79T^{2} \)
83 \( 1 - 2.09T + 83T^{2} \)
89 \( 1 - 3.39T + 89T^{2} \)
97 \( 1 + 2.39T + 97T^{2} \)
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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

−9.909690724418764979675476989283, −9.251484283982329155265377174552, −8.465157922028266551654679290940, −7.71628845206791363792837877103, −6.58684002998486052032557530845, −5.47174431640610847000982248692, −4.68743584837260494509493933048, −3.53785132501114809093491309714, −2.70717446011079999376010246326, −1.96632893080951077507603908848, 1.96632893080951077507603908848, 2.70717446011079999376010246326, 3.53785132501114809093491309714, 4.68743584837260494509493933048, 5.47174431640610847000982248692, 6.58684002998486052032557530845, 7.71628845206791363792837877103, 8.465157922028266551654679290940, 9.251484283982329155265377174552, 9.909690724418764979675476989283

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