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Properties

Label 2-931-133.132-c1-0-52
Degree $2$
Conductor $931$
Sign $0.455 + 0.890i$
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  + 2.17i·2-s + 0.445·3-s − 2.71·4-s − 3.02i·5-s + 0.967i·6-s − 1.55i·8-s − 2.80·9-s + 6.56·10-s − 1.10·11-s − 1.20·12-s − 0.222·13-s − 1.34i·15-s − 2.06·16-s − 3.86i·17-s − 6.08i·18-s + (−4.14 + 1.35i)19-s + ⋯
L(s)  = 1  + 1.53i·2-s + 0.257·3-s − 1.35·4-s − 1.35i·5-s + 0.394i·6-s − 0.548i·8-s − 0.933·9-s + 2.07·10-s − 0.333·11-s − 0.349·12-s − 0.0618·13-s − 0.347i·15-s − 0.515·16-s − 0.936i·17-s − 1.43i·18-s + (−0.950 + 0.310i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.455 + 0.890i$
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: \((2,\ 931,\ (\ :1/2),\ 0.455 + 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.427310 - 0.261341i\)
\(L(\frac12)\) \(\approx\) \(0.427310 - 0.261341i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.14 - 1.35i)T \)
good2 \( 1 - 2.17iT - 2T^{2} \)
3 \( 1 - 0.445T + 3T^{2} \)
5 \( 1 + 3.02iT - 5T^{2} \)
11 \( 1 + 1.10T + 11T^{2} \)
13 \( 1 + 0.222T + 13T^{2} \)
17 \( 1 + 3.86iT - 17T^{2} \)
23 \( 1 + 4.66T + 23T^{2} \)
29 \( 1 + 1.81iT - 29T^{2} \)
31 \( 1 + 2.29T + 31T^{2} \)
37 \( 1 + 9.03iT - 37T^{2} \)
41 \( 1 - 8.81T + 41T^{2} \)
43 \( 1 + 5.54T + 43T^{2} \)
47 \( 1 + 10.3iT - 47T^{2} \)
53 \( 1 - 12.2iT - 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 - 1.91iT - 61T^{2} \)
67 \( 1 + 7.11iT - 67T^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 - 2.61iT - 73T^{2} \)
79 \( 1 - 11.2iT - 79T^{2} \)
83 \( 1 - 0.147iT - 83T^{2} \)
89 \( 1 - 5.95T + 89T^{2} \)
97 \( 1 - 5.89T + 97T^{2} \)
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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

−9.150791999104960700590201901132, −9.068094267440345088498023232879, −8.034904461823667395966546283409, −7.66261070018158236414875692795, −6.37310166675145783063994535477, −5.62084332861351426186756511417, −4.96855393958597811557263689418, −4.02023982432218533618138225025, −2.29287446256686024429420034530, −0.20989486119530000261686871765, 1.91104039798978433712833593239, 2.78188478303956672146320991106, 3.42167458254110643783983925822, 4.45844561671812620420368333537, 5.93264104407791005957642511589, 6.72781019304291430080936434350, 7.943269772586789945924445689409, 8.735196275673593527917866731895, 9.771206935616748295112169495646, 10.40879956407410323931171355139

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