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Properties

Label 2-931-931.417-c0-0-0
Degree $2$
Conductor $931$
Sign $-0.918 + 0.394i$
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  + (0.826 − 0.563i)4-s + (−1.63 − 0.246i)5-s + (−0.988 − 0.149i)7-s + (−0.733 + 0.680i)9-s + (−1.40 − 1.29i)11-s + (0.365 − 0.930i)16-s + (−0.134 + 1.79i)17-s + (−0.5 − 0.866i)19-s + (−1.48 + 0.716i)20-s + (−0.109 − 1.46i)23-s + (1.65 + 0.510i)25-s + (−0.900 + 0.433i)28-s + (1.57 + 0.487i)35-s + (−0.222 + 0.974i)36-s + (0.0931 + 0.116i)43-s + (−1.88 − 0.284i)44-s + ⋯
L(s)  = 1  + (0.826 − 0.563i)4-s + (−1.63 − 0.246i)5-s + (−0.988 − 0.149i)7-s + (−0.733 + 0.680i)9-s + (−1.40 − 1.29i)11-s + (0.365 − 0.930i)16-s + (−0.134 + 1.79i)17-s + (−0.5 − 0.866i)19-s + (−1.48 + 0.716i)20-s + (−0.109 − 1.46i)23-s + (1.65 + 0.510i)25-s + (−0.900 + 0.433i)28-s + (1.57 + 0.487i)35-s + (−0.222 + 0.974i)36-s + (0.0931 + 0.116i)43-s + (−1.88 − 0.284i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $-0.918 + 0.394i$
Analytic conductor: \(0.464629\)
Root analytic conductor: \(0.681637\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :0),\ -0.918 + 0.394i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3270829405\)
\(L(\frac12)\) \(\approx\) \(0.3270829405\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.988 + 0.149i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.826 + 0.563i)T^{2} \)
3 \( 1 + (0.733 - 0.680i)T^{2} \)
5 \( 1 + (1.63 + 0.246i)T + (0.955 + 0.294i)T^{2} \)
11 \( 1 + (1.40 + 1.29i)T + (0.0747 + 0.997i)T^{2} \)
13 \( 1 + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (0.134 - 1.79i)T + (-0.988 - 0.149i)T^{2} \)
23 \( 1 + (0.109 + 1.46i)T + (-0.988 + 0.149i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.365 - 0.930i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (-0.0931 - 0.116i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.698 + 0.215i)T + (0.826 - 0.563i)T^{2} \)
53 \( 1 + (-0.365 + 0.930i)T^{2} \)
59 \( 1 + (-0.955 + 0.294i)T^{2} \)
61 \( 1 + (1.21 + 0.825i)T + (0.365 + 0.930i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (0.955 + 0.294i)T + (0.826 + 0.563i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.0332 + 0.145i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.0747 + 0.997i)T^{2} \)
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

−10.36964802931660956946299724660, −8.725026972727036140748664661591, −8.248157978388322175791794536417, −7.47491059180326357423503728637, −6.41387630304121533829432492091, −5.71337068434517124819701419346, −4.54416361365852207518825872182, −3.34094963509337701320773502815, −2.56166807712821985441792158982, −0.28453940898416654397080149977, 2.60488781548703194351160334150, 3.22903713977861968459193595555, 4.15221895815417283614898126943, 5.48003308635139232281967532420, 6.66383809408964407547332863656, 7.46560372425716084302212046136, 7.71895924374639848301986100956, 8.875058697475451043159324591934, 9.857537287753525853277624122604, 10.77466600464092741885386467000

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