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Properties

Label 2-931-133.69-c1-0-7
Degree $2$
Conductor $931$
Sign $-0.992 + 0.120i$
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  + (−0.395 + 0.228i)2-s + (0.895 + 1.55i)3-s + (−0.895 + 1.55i)4-s + (1.10 − 0.637i)5-s + (−0.708 − 0.409i)6-s − 1.73i·8-s + (−0.104 + 0.180i)9-s + (−0.291 + 0.504i)10-s − 3·11-s − 3.20·12-s + (−1.89 + 3.28i)13-s + (1.97 + 1.14i)15-s + (−1.39 − 2.41i)16-s + (−3.79 + 2.18i)17-s − 0.0953i·18-s + (−3.5 − 2.59i)19-s + ⋯
L(s)  = 1  + (−0.279 + 0.161i)2-s + (0.517 + 0.895i)3-s + (−0.447 + 0.775i)4-s + (0.493 − 0.285i)5-s + (−0.289 − 0.167i)6-s − 0.612i·8-s + (−0.0347 + 0.0602i)9-s + (−0.0921 + 0.159i)10-s − 0.904·11-s − 0.926·12-s + (−0.525 + 0.910i)13-s + (0.510 + 0.294i)15-s + (−0.348 − 0.604i)16-s + (−0.919 + 0.530i)17-s − 0.0224i·18-s + (−0.802 − 0.596i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $-0.992 + 0.120i$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (734, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ -0.992 + 0.120i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0486052 - 0.804946i\)
\(L(\frac12)\) \(\approx\) \(0.0486052 - 0.804946i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 + 2.59i)T \)
good2 \( 1 + (0.395 - 0.228i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.895 - 1.55i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.10 + 0.637i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (1.89 - 3.28i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.79 - 2.18i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.79 - 6.56i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.08 + 1.77i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 10.0iT - 37T^{2} \)
41 \( 1 + (2.68 + 4.65i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.39 - 5.88i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.29 - 3.05i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.10 + 0.637i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.89 + 8.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.18 - 0.685i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3 - 1.73i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.08 - 1.77i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.37 + 3.10i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6 + 3.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.5iT - 83T^{2} \)
89 \( 1 + (1.18 - 2.05i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.12701356991643340799461244567, −9.476125681441553167702448502613, −9.014721786335137714897076243365, −8.195903489885326560532769835399, −7.32948167688346905256962950171, −6.27643571532440645242247202306, −4.93902173643255896992097837094, −4.27805183827116176604429264291, −3.36701359749618900126840458929, −2.12049490918363799560122862734, 0.36230585079996246539337598353, 2.08654832906852039557501557596, 2.49245545550265411218034805669, 4.33810398012366406332827094363, 5.38456500669491030898734788026, 6.19389072862872179643114891722, 7.18342898208804996754083040768, 8.088323412962795375890509888108, 8.676018720191763967356492238348, 9.711788631161298037669037427232

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