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Properties

Label 2-931-133.132-c1-0-62
Degree $2$
Conductor $931$
Sign $0.0543 - 0.998i$
Analytic cond. $7.43407$
Root an. cond. $2.72654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.46i·2-s + 1.27·3-s − 4.07·4-s − 0.595i·5-s − 3.14i·6-s + 5.10i·8-s − 1.37·9-s − 1.46·10-s − 5.30·11-s − 5.19·12-s − 0.996·13-s − 0.759i·15-s + 4.43·16-s + 1.72i·17-s + 3.38i·18-s + (−1.56 − 4.06i)19-s + ⋯
L(s)  = 1  − 1.74i·2-s + 0.736·3-s − 2.03·4-s − 0.266i·5-s − 1.28i·6-s + 1.80i·8-s − 0.457·9-s − 0.463·10-s − 1.59·11-s − 1.49·12-s − 0.276·13-s − 0.196i·15-s + 1.10·16-s + 0.417i·17-s + 0.797i·18-s + (−0.358 − 0.933i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0543 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0543 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.0543 - 0.998i$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (930, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ 0.0543 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.378125 + 0.358109i\)
\(L(\frac12)\) \(\approx\) \(0.378125 + 0.358109i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.56 + 4.06i)T \)
good2 \( 1 + 2.46iT - 2T^{2} \)
3 \( 1 - 1.27T + 3T^{2} \)
5 \( 1 + 0.595iT - 5T^{2} \)
11 \( 1 + 5.30T + 11T^{2} \)
13 \( 1 + 0.996T + 13T^{2} \)
17 \( 1 - 1.72iT - 17T^{2} \)
23 \( 1 - 2.64T + 23T^{2} \)
29 \( 1 - 0.0820iT - 29T^{2} \)
31 \( 1 + 7.37T + 31T^{2} \)
37 \( 1 + 0.771iT - 37T^{2} \)
41 \( 1 + 7.80T + 41T^{2} \)
43 \( 1 + 5.74T + 43T^{2} \)
47 \( 1 + 8.93iT - 47T^{2} \)
53 \( 1 - 9.55iT - 53T^{2} \)
59 \( 1 - 12.2T + 59T^{2} \)
61 \( 1 - 2.88iT - 61T^{2} \)
67 \( 1 + 1.01iT - 67T^{2} \)
71 \( 1 + 14.9iT - 71T^{2} \)
73 \( 1 + 6.79iT - 73T^{2} \)
79 \( 1 - 11.5iT - 79T^{2} \)
83 \( 1 + 15.0iT - 83T^{2} \)
89 \( 1 - 13.9T + 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−9.533005334252637874076518432312, −8.782047365228813961606305022083, −8.280797339700258106202712587944, −7.12544426458171026052540995070, −5.42401147087888623064339549163, −4.74289983818796872228811305081, −3.48212395277507755174044655484, −2.77266288531446522390168239652, −1.95659187639228820200433287968, −0.20706635390275744905494709513, 2.49833744765039365837273484517, 3.62183880780860632530851494596, 5.04378859395030944971008690165, 5.48043452807319840194831253585, 6.62417300843629071940003476565, 7.39362375436127077768577347454, 8.124204864386438370051509347400, 8.582302145302478399817377570176, 9.514506507710535438008371452140, 10.36986734079395645588518592023

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