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Properties

Label 2-931-931.170-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  + (−0.988 − 0.149i)4-s + (1.44 + 1.34i)5-s + (−0.733 − 0.680i)7-s + (0.826 − 0.563i)9-s + (0.123 + 0.0841i)11-s + (0.955 + 0.294i)16-s + (0.455 − 1.16i)17-s + (−0.5 + 0.866i)19-s + (−1.23 − 1.54i)20-s + (0.603 + 1.53i)23-s + (0.217 + 2.90i)25-s + (0.623 + 0.781i)28-s + (−0.147 − 1.97i)35-s + (−0.900 + 0.433i)36-s + (−0.162 − 0.712i)43-s + (−0.109 − 0.101i)44-s + ⋯
L(s)  = 1  + (−0.988 − 0.149i)4-s + (1.44 + 1.34i)5-s + (−0.733 − 0.680i)7-s + (0.826 − 0.563i)9-s + (0.123 + 0.0841i)11-s + (0.955 + 0.294i)16-s + (0.455 − 1.16i)17-s + (−0.5 + 0.866i)19-s + (−1.23 − 1.54i)20-s + (0.603 + 1.53i)23-s + (0.217 + 2.90i)25-s + (0.623 + 0.781i)28-s + (−0.147 − 1.97i)35-s + (−0.900 + 0.433i)36-s + (−0.162 − 0.712i)43-s + (−0.109 − 0.101i)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: $\chi_{931} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :0),\ 0.926 - 0.375i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.025524963\)
\(L(\frac12)\) \(\approx\) \(1.025524963\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.733 + 0.680i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.988 + 0.149i)T^{2} \)
3 \( 1 + (-0.826 + 0.563i)T^{2} \)
5 \( 1 + (-1.44 - 1.34i)T + (0.0747 + 0.997i)T^{2} \)
11 \( 1 + (-0.123 - 0.0841i)T + (0.365 + 0.930i)T^{2} \)
13 \( 1 + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + (-0.455 + 1.16i)T + (-0.733 - 0.680i)T^{2} \)
23 \( 1 + (-0.603 - 1.53i)T + (-0.733 + 0.680i)T^{2} \)
29 \( 1 + (0.222 - 0.974i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.955 + 0.294i)T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.162 + 0.712i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.142 + 1.90i)T + (-0.988 - 0.149i)T^{2} \)
53 \( 1 + (-0.955 - 0.294i)T^{2} \)
59 \( 1 + (-0.0747 + 0.997i)T^{2} \)
61 \( 1 + (1.63 - 0.246i)T + (0.955 - 0.294i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (0.0747 + 0.997i)T + (-0.988 + 0.149i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.658 + 0.317i)T + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.365 + 0.930i)T^{2} \)
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

−10.13866978369385169826862810728, −9.615092264538451722839536197799, −9.142417561229839719495277405417, −7.49397470896204494000355645030, −6.90785547323479168018654584112, −6.04751630289246307695254806577, −5.23283365094573642316296901570, −3.83667847492176705992645117377, −3.12948205196622075635613706914, −1.52544468027691354541971054399, 1.31187944191465592582911311381, 2.62596471999100069192300282126, 4.30623756165530354724131300528, 4.88768058055910615629156637391, 5.79435727592644947222231758852, 6.50821456889866417486892843392, 8.088972562575428151330374389274, 8.767723465455900618096234660153, 9.318211764949507953728299781995, 9.960405421832515209238870897673

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