↑

Properties

Label 2-31e2-31.19-c1-0-41
Degree $2$
Conductor $961$
Sign $-0.293 + 0.955i$
Analytic cond. $7.67362$
Root an. cond. $2.77013$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)2-s + (−2.58 − 1.15i)3-s + (−0.309 − 0.951i)4-s + (−1.41 − 2.44i)6-s + (2.67 − 2.97i)7-s + (0.927 − 2.85i)8-s + (3.34 + 3.71i)9-s + (2.76 + 0.588i)11-s + (−0.295 + 2.81i)12-s + (0.147 + 1.40i)13-s + (3.91 − 0.831i)14-s + (0.809 − 0.587i)16-s + (1.38 − 0.294i)17-s + (0.522 + 4.97i)18-s + (0.418 − 3.97i)19-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−1.49 − 0.664i)3-s + (−0.154 − 0.475i)4-s + (−0.577 − 0.999i)6-s + (1.01 − 1.12i)7-s + (0.327 − 1.00i)8-s + (1.11 + 1.23i)9-s + (0.834 + 0.177i)11-s + (−0.0853 + 0.812i)12-s + (0.0409 + 0.390i)13-s + (1.04 − 0.222i)14-s + (0.202 − 0.146i)16-s + (0.335 − 0.0713i)17-s + (0.123 + 1.17i)18-s + (0.0959 − 0.912i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.293 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.293 + 0.955i$
Analytic conductor: \(7.67362\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (732, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :1/2),\ -0.293 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.812013 - 1.09863i\)
\(L(\frac12)\) \(\approx\) \(0.812013 - 1.09863i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (2.58 + 1.15i)T + (2.00 + 2.22i)T^{2} \)
5 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.67 + 2.97i)T + (-0.731 - 6.96i)T^{2} \)
11 \( 1 + (-2.76 - 0.588i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-0.147 - 1.40i)T + (-12.7 + 2.70i)T^{2} \)
17 \( 1 + (-1.38 + 0.294i)T + (15.5 - 6.91i)T^{2} \)
19 \( 1 + (-0.418 + 3.97i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (-1.74 + 5.37i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (1.14 + 0.831i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + (2.12 + 3.67i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.82 - 0.813i)T + (27.4 - 30.4i)T^{2} \)
43 \( 1 + (-0.886 + 8.43i)T + (-42.0 - 8.94i)T^{2} \)
47 \( 1 + (9.70 - 7.05i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.83 + 3.15i)T + (-5.54 + 52.7i)T^{2} \)
59 \( 1 + (-7.30 - 3.25i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + 1.41T + 61T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.35 - 5.94i)T + (-7.42 + 70.6i)T^{2} \)
73 \( 1 + (4.14 + 0.882i)T + (66.6 + 29.6i)T^{2} \)
79 \( 1 + (11.0 - 2.35i)T + (72.1 - 32.1i)T^{2} \)
83 \( 1 + (-12.9 + 5.75i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (-2.18 - 6.72i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-2.47 - 7.60i)T + (-78.4 + 57.0i)T^{2} \)
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

−10.07038699316510370719404635145, −8.972464848928178742120070038000, −7.57895374479501076626564667187, −6.88130728714059573362793286384, −6.47768870958118025097483838960, −5.30859811478314284811703005117, −4.81058598274686008896694619709, −3.97676743722778700258254794733, −1.56619074760910302975738663497, −0.72945715869650744164688546973, 1.60435201015809476956508550983, 3.21399022082529015332027466035, 4.23623117102496953111139441380, 5.10613386201014919433369841728, 5.54145138931704030075699310536, 6.50140282056551129464779003650, 7.88277045882834963763488861456, 8.618075183535044617338208418060, 9.629822643272648922506309784119, 10.58096688933377270693953414828

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