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Properties

Label 2-931-931.911-c0-0-0
Degree $2$
Conductor $931$
Sign $0.926 + 0.375i$
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  + (0.623 + 0.781i)4-s + (−0.277 − 1.21i)5-s + (−0.222 − 0.974i)7-s + (−0.900 + 0.433i)9-s + (1.62 + 0.781i)11-s + (−0.222 + 0.974i)16-s + (0.777 − 0.974i)17-s + 19-s + (0.777 − 0.974i)20-s + (−1.12 − 1.40i)23-s + (−0.499 + 0.240i)25-s + (0.623 − 0.781i)28-s + (−1.12 + 0.541i)35-s + (−0.900 − 0.433i)36-s + (−0.277 + 1.21i)43-s + (0.400 + 1.75i)44-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)4-s + (−0.277 − 1.21i)5-s + (−0.222 − 0.974i)7-s + (−0.900 + 0.433i)9-s + (1.62 + 0.781i)11-s + (−0.222 + 0.974i)16-s + (0.777 − 0.974i)17-s + 19-s + (0.777 − 0.974i)20-s + (−1.12 − 1.40i)23-s + (−0.499 + 0.240i)25-s + (0.623 − 0.781i)28-s + (−1.12 + 0.541i)35-s + (−0.900 − 0.433i)36-s + (−0.277 + 1.21i)43-s + (0.400 + 1.75i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.926 + 0.375i$
Analytic conductor: \(0.464629\)
Root analytic conductor: \(0.681637\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (911, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :0),\ 0.926 + 0.375i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.106345142\)
\(L(\frac12)\) \(\approx\) \(1.106345142\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.222 + 0.974i)T \)
19 \( 1 - T \)
good2 \( 1 + (-0.623 - 0.781i)T^{2} \)
3 \( 1 + (0.900 - 0.433i)T^{2} \)
5 \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \)
11 \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \)
13 \( 1 + (-0.623 - 0.781i)T^{2} \)
17 \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \)
23 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.222 + 0.974i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
47 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.222 - 0.974i)T^{2} \)
59 \( 1 + (0.900 + 0.433i)T^{2} \)
61 \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (1.80 - 0.867i)T + (0.623 - 0.781i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.623 + 0.781i)T^{2} \)
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

−10.14207113041658605769745559224, −9.280665427695393081638087165650, −8.478461456339516786094478291194, −7.69916733950563444249251886663, −6.99971228722949750607831925270, −5.99636659216645356066428245282, −4.67308535975783750673509298121, −4.02943755576197296745044464077, −2.88277033298563192734525716444, −1.29894061915536214363097788860, 1.67638840875428999191867553867, 3.07873248992556478261823086309, 3.59576005824425631321879507489, 5.62156107668167553113355212515, 5.98586777351614792251493156499, 6.67832659865677979787040057670, 7.67559585962806123049328767210, 8.812001010068893290496575126989, 9.536668067556840121183789957190, 10.34837344059659749193891339121

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