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Properties

Label 2-931-931.645-c0-0-0
Degree $2$
Conductor $931$
Sign $0.518 - 0.855i$
Analytic cond. $0.464629$
Root an. cond. $0.681637$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.222 + 0.974i)4-s + (0.400 − 0.193i)5-s + (−0.900 + 0.433i)7-s + (0.623 − 0.781i)9-s + (0.777 + 0.974i)11-s + (−0.900 − 0.433i)16-s + (0.0990 + 0.433i)17-s + 19-s + (0.0990 + 0.433i)20-s + (−0.277 + 1.21i)23-s + (−0.499 + 0.626i)25-s + (−0.222 − 0.974i)28-s + (−0.277 + 0.347i)35-s + (0.623 + 0.781i)36-s + (0.400 + 0.193i)43-s + (−1.12 + 0.541i)44-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)4-s + (0.400 − 0.193i)5-s + (−0.900 + 0.433i)7-s + (0.623 − 0.781i)9-s + (0.777 + 0.974i)11-s + (−0.900 − 0.433i)16-s + (0.0990 + 0.433i)17-s + 19-s + (0.0990 + 0.433i)20-s + (−0.277 + 1.21i)23-s + (−0.499 + 0.626i)25-s + (−0.222 − 0.974i)28-s + (−0.277 + 0.347i)35-s + (0.623 + 0.781i)36-s + (0.400 + 0.193i)43-s + (−1.12 + 0.541i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.518 - 0.855i$
Analytic conductor: \(0.464629\)
Root analytic conductor: \(0.681637\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (645, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :0),\ 0.518 - 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9881625314\)
\(L(\frac12)\) \(\approx\) \(0.9881625314\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.900 - 0.433i)T \)
19 \( 1 - T \)
good2 \( 1 + (0.222 - 0.974i)T^{2} \)
3 \( 1 + (-0.623 + 0.781i)T^{2} \)
5 \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \)
11 \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \)
13 \( 1 + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \)
23 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.900 - 0.433i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.900 + 0.433i)T^{2} \)
59 \( 1 + (-0.623 - 0.781i)T^{2} \)
61 \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (-1.24 + 1.56i)T + (-0.222 - 0.974i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.222 + 0.974i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.980255602387274321372386518571, −9.492790845000928710832552255621, −9.032452225044242792277688825378, −7.78973048327571910098886726176, −7.03502336539441146093851343395, −6.27130517147232129795634013108, −5.11969006606724903286342298946, −3.88397489654458977986119203932, −3.32700310992845950264616727832, −1.79154570611984243984590297293, 1.09517955400299238498098361360, 2.57410783676967902633252719867, 3.90142597549827395726344795092, 4.89969812722427895110521009746, 5.97452408866385219411152681625, 6.50164596628220650945125139797, 7.46904564904610205254178040413, 8.649249158720430704977640650139, 9.568306487665927830297017681511, 10.04413930567429243805745832113

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