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Properties

Label 2-931-931.683-c0-0-0
Degree $2$
Conductor $931$
Sign $-0.999 - 0.0213i$
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  + (−0.733 − 0.680i)4-s + (−1.21 − 0.825i)5-s + (0.826 + 0.563i)7-s + (−0.988 − 0.149i)9-s + (−0.722 + 0.108i)11-s + (0.0747 + 0.997i)16-s + (−0.425 − 0.131i)17-s + (−0.5 − 0.866i)19-s + (0.326 + 1.42i)20-s + (−1.88 + 0.582i)23-s + (0.419 + 1.07i)25-s + (−0.222 − 0.974i)28-s + (−0.535 − 1.36i)35-s + (0.623 + 0.781i)36-s + (−1.72 − 0.829i)43-s + (0.603 + 0.411i)44-s + ⋯
L(s)  = 1  + (−0.733 − 0.680i)4-s + (−1.21 − 0.825i)5-s + (0.826 + 0.563i)7-s + (−0.988 − 0.149i)9-s + (−0.722 + 0.108i)11-s + (0.0747 + 0.997i)16-s + (−0.425 − 0.131i)17-s + (−0.5 − 0.866i)19-s + (0.326 + 1.42i)20-s + (−1.88 + 0.582i)23-s + (0.419 + 1.07i)25-s + (−0.222 − 0.974i)28-s + (−0.535 − 1.36i)35-s + (0.623 + 0.781i)36-s + (−1.72 − 0.829i)43-s + (0.603 + 0.411i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $-0.999 - 0.0213i$
Analytic conductor: \(0.464629\)
Root analytic conductor: \(0.681637\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :0),\ -0.999 - 0.0213i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1985774303\)
\(L(\frac12)\) \(\approx\) \(0.1985774303\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.826 - 0.563i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.733 + 0.680i)T^{2} \)
3 \( 1 + (0.988 + 0.149i)T^{2} \)
5 \( 1 + (1.21 + 0.825i)T + (0.365 + 0.930i)T^{2} \)
11 \( 1 + (0.722 - 0.108i)T + (0.955 - 0.294i)T^{2} \)
13 \( 1 + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (0.425 + 0.131i)T + (0.826 + 0.563i)T^{2} \)
23 \( 1 + (1.88 - 0.582i)T + (0.826 - 0.563i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.0747 + 0.997i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (1.72 + 0.829i)T + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (-0.0546 + 0.139i)T + (-0.733 - 0.680i)T^{2} \)
53 \( 1 + (-0.0747 - 0.997i)T^{2} \)
59 \( 1 + (-0.365 + 0.930i)T^{2} \)
61 \( 1 + (-1.44 + 1.34i)T + (0.0747 - 0.997i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.365 + 0.930i)T + (-0.733 + 0.680i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1.19 + 1.49i)T + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.955 - 0.294i)T^{2} \)
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

−9.732087294908766067997271550066, −8.661675127617316266505389972930, −8.476558341202800341277024506695, −7.65835270350537062331337056112, −6.17147120956769813072530685360, −5.20757338281396887488383977685, −4.71953341722844772416745047466, −3.69735841690031940052869474345, −2.06133405217890040743709606015, −0.18701152606822169589848569574, 2.50281268598276109973769509739, 3.65680033292007073119198876218, 4.23682673789021277886961340256, 5.31249164048442737478808929992, 6.58307418154629725343308895501, 7.73791463763158302758704186582, 8.074509781332394271320081207163, 8.581651887887075172736041884997, 10.05784339932992898018125731056, 10.72910136517964334733247567023

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