↑

Properties

Label 2-931-931.436-c0-0-0
Degree $2$
Conductor $931$
Sign $0.481 + 0.876i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.955 − 0.294i)4-s + (0.142 − 1.90i)5-s + (0.0747 − 0.997i)7-s + (0.365 + 0.930i)9-s + (−0.722 + 1.84i)11-s + (0.826 − 0.563i)16-s + (0.326 − 0.302i)17-s + (−0.5 + 0.866i)19-s + (−0.425 − 1.86i)20-s + (−0.535 − 0.496i)23-s + (−2.62 − 0.395i)25-s + (−0.222 − 0.974i)28-s + (−1.88 − 0.284i)35-s + (0.623 + 0.781i)36-s + (1.32 + 0.636i)43-s + (−0.147 + 1.97i)44-s + ⋯
L(s)  = 1  + (0.955 − 0.294i)4-s + (0.142 − 1.90i)5-s + (0.0747 − 0.997i)7-s + (0.365 + 0.930i)9-s + (−0.722 + 1.84i)11-s + (0.826 − 0.563i)16-s + (0.326 − 0.302i)17-s + (−0.5 + 0.866i)19-s + (−0.425 − 1.86i)20-s + (−0.535 − 0.496i)23-s + (−2.62 − 0.395i)25-s + (−0.222 − 0.974i)28-s + (−1.88 − 0.284i)35-s + (0.623 + 0.781i)36-s + (1.32 + 0.636i)43-s + (−0.147 + 1.97i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.481 + 0.876i$
Analytic conductor: \(0.464629\)
Root analytic conductor: \(0.681637\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (436, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :0),\ 0.481 + 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.244847265\)
\(L(\frac12)\) \(\approx\) \(1.244847265\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.0747 + 0.997i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.955 + 0.294i)T^{2} \)
3 \( 1 + (-0.365 - 0.930i)T^{2} \)
5 \( 1 + (-0.142 + 1.90i)T + (-0.988 - 0.149i)T^{2} \)
11 \( 1 + (0.722 - 1.84i)T + (-0.733 - 0.680i)T^{2} \)
13 \( 1 + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.326 + 0.302i)T + (0.0747 - 0.997i)T^{2} \)
23 \( 1 + (0.535 + 0.496i)T + (0.0747 + 0.997i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.826 - 0.563i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (-1.32 - 0.636i)T + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (1.63 - 0.246i)T + (0.955 - 0.294i)T^{2} \)
53 \( 1 + (-0.826 + 0.563i)T^{2} \)
59 \( 1 + (0.988 - 0.149i)T^{2} \)
61 \( 1 + (-0.698 - 0.215i)T + (0.826 + 0.563i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (-0.988 - 0.149i)T + (0.955 + 0.294i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.914 - 1.14i)T + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.733 - 0.680i)T^{2} \)
97 \( 1 - T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.00998974299057213813340430827, −9.640550434923029321259604974228, −8.077315066411742335466464989295, −7.80991362831868491878477835764, −6.86408239301547854348197130832, −5.58711090708542263541373768980, −4.80143418963005399798868329223, −4.20550379886040510390486990879, −2.18147832742595872276909544889, −1.41498514178854086254503295525, 2.20921680240799776125907303548, 3.05578402586603668746804653317, 3.56510916960175904583355203195, 5.72208223500396974406671604544, 6.17333753764184944769059383947, 6.86263496252385146540491260706, 7.77788784388352566434682172579, 8.613686116832009583081906777426, 9.780251030726493296002586335987, 10.62074879254784528359023380258

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