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Properties

Label 2-931-133.122-c1-0-50
Degree $2$
Conductor $931$
Sign $-0.189 + 0.981i$
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  − 0.525i·2-s + (0.0476 + 0.0825i)3-s + 1.72·4-s − 3.78i·5-s + (0.0434 − 0.0250i)6-s − 1.95i·8-s + (1.49 − 2.59i)9-s − 1.98·10-s + (0.250 − 0.433i)11-s + (0.0821 + 0.142i)12-s + (3.23 + 5.60i)13-s + (0.312 − 0.180i)15-s + 2.41·16-s + (3.44 − 1.99i)17-s + (−1.36 − 0.786i)18-s + (−4.16 + 1.28i)19-s + ⋯
L(s)  = 1  − 0.371i·2-s + (0.0275 + 0.0476i)3-s + 0.861·4-s − 1.69i·5-s + (0.0177 − 0.0102i)6-s − 0.692i·8-s + (0.498 − 0.863i)9-s − 0.629·10-s + (0.0754 − 0.130i)11-s + (0.0237 + 0.0410i)12-s + (0.897 + 1.55i)13-s + (0.0806 − 0.0465i)15-s + 0.604·16-s + (0.836 − 0.482i)17-s + (−0.321 − 0.185i)18-s + (−0.955 + 0.294i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $-0.189 + 0.981i$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ -0.189 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33948 - 1.62261i\)
\(L(\frac12)\) \(\approx\) \(1.33948 - 1.62261i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.16 - 1.28i)T \)
good2 \( 1 + 0.525iT - 2T^{2} \)
3 \( 1 + (-0.0476 - 0.0825i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 3.78iT - 5T^{2} \)
11 \( 1 + (-0.250 + 0.433i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.23 - 5.60i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.44 + 1.99i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.16 + 2.01i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.67 - 3.27i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.24 - 3.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.502 - 0.289i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.62 + 4.54i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.670 + 1.16i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.37 + 3.67i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.25iT - 53T^{2} \)
59 \( 1 + (-2.81 - 4.88i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.27 - 0.737i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + 5.62iT - 67T^{2} \)
71 \( 1 + (-5.27 - 3.04i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.27 + 3.62i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 8.97iT - 79T^{2} \)
83 \( 1 - 8.36iT - 83T^{2} \)
89 \( 1 + (-1.78 + 3.09i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.29 - 2.24i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.657190130412224926347142802305, −9.093233250301307677567379752456, −8.397561032733056128088596040115, −7.19849192959000384657406877364, −6.41766662614541343432782162265, −5.46646190561369902776563871351, −4.26946697995452100734113289114, −3.60259882611829908398000770645, −1.85535174052726675386089168604, −1.06199936879856887752257204748, 1.88897856872438313470047370405, 2.88033603384139626119303709033, 3.73508714900743428551350233497, 5.43690524801491556800498761062, 6.15594048396180735584256900387, 6.89235380271283106871744555387, 7.82609472430198703293591049951, 8.025337588329023934215364563687, 9.786525477337197040265192294137, 10.53468579192216811681800556083

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