Properties

Label 2-931-931.303-c0-0-0
Degree $2$
Conductor $931$
Sign $0.117 - 0.993i$
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  + (0.0747 + 0.997i)4-s + (0.0546 + 0.139i)5-s + (0.365 + 0.930i)7-s + (0.955 + 0.294i)9-s + (−1.40 + 0.432i)11-s + (−0.988 + 0.149i)16-s + (−1.48 − 1.01i)17-s + (−0.5 + 0.866i)19-s + (−0.134 + 0.0648i)20-s + (1.57 − 1.07i)23-s + (0.716 − 0.664i)25-s + (−0.900 + 0.433i)28-s + (−0.109 + 0.101i)35-s + (−0.222 + 0.974i)36-s + (1.03 + 1.29i)43-s + (−0.535 − 1.36i)44-s + ⋯
L(s)  = 1  + (0.0747 + 0.997i)4-s + (0.0546 + 0.139i)5-s + (0.365 + 0.930i)7-s + (0.955 + 0.294i)9-s + (−1.40 + 0.432i)11-s + (−0.988 + 0.149i)16-s + (−1.48 − 1.01i)17-s + (−0.5 + 0.866i)19-s + (−0.134 + 0.0648i)20-s + (1.57 − 1.07i)23-s + (0.716 − 0.664i)25-s + (−0.900 + 0.433i)28-s + (−0.109 + 0.101i)35-s + (−0.222 + 0.974i)36-s + (1.03 + 1.29i)43-s + (−0.535 − 1.36i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.117 - 0.993i$
Analytic conductor: \(0.464629\)
Root analytic conductor: \(0.681637\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :0),\ 0.117 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.008430904\)
\(L(\frac12)\) \(\approx\) \(1.008430904\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.365 - 0.930i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.0747 - 0.997i)T^{2} \)
3 \( 1 + (-0.955 - 0.294i)T^{2} \)
5 \( 1 + (-0.0546 - 0.139i)T + (-0.733 + 0.680i)T^{2} \)
11 \( 1 + (1.40 - 0.432i)T + (0.826 - 0.563i)T^{2} \)
13 \( 1 + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (1.48 + 1.01i)T + (0.365 + 0.930i)T^{2} \)
23 \( 1 + (-1.57 + 1.07i)T + (0.365 - 0.930i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.988 + 0.149i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (-1.03 - 1.29i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (-1.44 - 1.34i)T + (0.0747 + 0.997i)T^{2} \)
53 \( 1 + (0.988 - 0.149i)T^{2} \)
59 \( 1 + (0.733 + 0.680i)T^{2} \)
61 \( 1 + (-0.142 + 1.90i)T + (-0.988 - 0.149i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.733 + 0.680i)T + (0.0747 - 0.997i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.367 + 1.61i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.826 - 0.563i)T^{2} \)
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

−10.72174149266425635228336440325, −9.473854677991206266340143710479, −8.718245549403540652277460557908, −7.928176382077720567182558715310, −7.21341937229161831228118988772, −6.35204150823558704321042041821, −4.86734385978402774408406636654, −4.50567497067772240975691211077, −2.81174973857918583803341830656, −2.27404570499574768330633283635, 1.04169122193521846165461187052, 2.34799260557309966540538935776, 3.96144587941993442990663578473, 4.84848924788025128965622908315, 5.59857308213869688502557911956, 6.91195736322121498269771147290, 7.20173088554037290871365562346, 8.573180774915044983322099909071, 9.257681771873502785798344678269, 10.36578395269350878119750323101

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