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Properties

Label 2-31e2-31.16-c1-0-39
Degree $2$
Conductor $961$
Sign $0.696 - 0.717i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)2-s + (2.28 + 1.66i)3-s + (−0.309 + 0.951i)4-s + 2.82·6-s + (1.23 − 3.80i)7-s + (0.927 + 2.85i)8-s + (1.54 + 4.75i)9-s + (−0.874 + 2.68i)11-s + (−2.28 + 1.66i)12-s + (1.14 + 0.831i)13-s + (−1.23 − 3.80i)14-s + (0.809 + 0.587i)16-s + (−0.437 − 1.34i)17-s + (4.04 + 2.93i)18-s + (3.23 − 2.35i)19-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (1.32 + 0.959i)3-s + (−0.154 + 0.475i)4-s + 1.15·6-s + (0.467 − 1.43i)7-s + (0.327 + 1.00i)8-s + (0.515 + 1.58i)9-s + (−0.263 + 0.811i)11-s + (−0.660 + 0.479i)12-s + (0.317 + 0.230i)13-s + (−0.330 − 1.01i)14-s + (0.202 + 0.146i)16-s + (−0.105 − 0.326i)17-s + (0.953 + 0.692i)18-s + (0.742 − 0.539i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.696 - 0.717i$
Analytic conductor: \(7.67362\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :1/2),\ 0.696 - 0.717i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.06050 + 1.29554i\)
\(L(\frac12)\) \(\approx\) \(3.06050 + 1.29554i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.28 - 1.66i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 + (-1.23 + 3.80i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (0.874 - 2.68i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.14 - 0.831i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.437 + 1.34i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.23 + 2.35i)T + (5.87 - 18.0i)T^{2} \)
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 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 + (-1.61 + 1.17i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-6.86 + 4.98i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (9.70 + 7.05i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.31 + 4.03i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.47 + 4.70i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 1.41T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (-2.47 - 7.60i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.31 + 4.03i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.49 - 10.7i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (11.4 - 8.31i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-2.18 + 6.72i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-2.47 + 7.60i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.973538457741944584899075283971, −9.456542974500367003488807838364, −8.473555284978486689319950831527, −7.65404638109471015533330562658, −7.22689560205712922827832843839, −5.21346479651623890150686004070, −4.42438464802039533043708033997, −3.85189044362347456404444368818, −3.08903106456124521169085803083, −1.89654529688565128923031207327, 1.31113221548804069886282760625, 2.45206236445472566432441286515, 3.38895091248599708654057576514, 4.70459820410866436362668442383, 5.91493333197006132677696812781, 6.20650893802327818171339240025, 7.59356736467406730993444258708, 8.134009340486775372294123363965, 8.957808401426216296612891557754, 9.521362785936465617989696723439

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