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Properties

Label 2-31e2-31.25-c1-0-21
Degree $2$
Conductor $961$
Sign $-0.0336 - 0.999i$
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  − 0.618·2-s + (0.618 + 1.07i)3-s − 1.61·4-s + (−0.5 + 0.866i)5-s + (−0.381 − 0.661i)6-s + (2.11 + 3.66i)7-s + 2.23·8-s + (0.736 − 1.27i)9-s + (0.309 − 0.535i)10-s + (1 − 1.73i)11-s + (−1.00 − 1.73i)12-s + (0.618 − 1.07i)13-s + (−1.30 − 2.26i)14-s − 1.23·15-s + 1.85·16-s + (2.61 + 4.53i)17-s + ⋯
L(s)  = 1  − 0.437·2-s + (0.356 + 0.618i)3-s − 0.809·4-s + (−0.223 + 0.387i)5-s + (−0.155 − 0.270i)6-s + (0.800 + 1.38i)7-s + 0.790·8-s + (0.245 − 0.424i)9-s + (0.0977 − 0.169i)10-s + (0.301 − 0.522i)11-s + (−0.288 − 0.499i)12-s + (0.171 − 0.296i)13-s + (−0.349 − 0.605i)14-s − 0.319·15-s + 0.463·16-s + (0.634 + 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0336 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0336 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.0336 - 0.999i$
Analytic conductor: \(7.67362\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :1/2),\ -0.0336 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.905019 + 0.936047i\)
\(L(\frac12)\) \(\approx\) \(0.905019 + 0.936047i\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.618T + 2T^{2} \)
3 \( 1 + (-0.618 - 1.07i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.11 - 3.66i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.618 + 1.07i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.61 - 4.53i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.11 + 1.93i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.70T + 23T^{2} \)
29 \( 1 + 7.23T + 29T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.5 - 6.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.61 + 2.80i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.47T + 47T^{2} \)
53 \( 1 + (0.763 - 1.32i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.11 - 1.93i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.59 - 11.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.236 - 0.408i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.854 - 1.47i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.47 + 2.54i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 1.70T + 89T^{2} \)
97 \( 1 - 1.94T + 97T^{2} \)
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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

−10.05780451480325125279168365589, −9.186240098364671365443879879439, −8.728432264237669532352399953226, −8.158748785665919726554886920512, −7.00138496968985407697684982333, −5.70456297078567516828990443589, −5.00891355527902292311227979055, −3.89513497647238750584806458222, −3.05880728431407142631392854060, −1.39475317742564480481464717283, 0.832873558040312700808757975265, 1.73220824648083177120416952496, 3.58167076977337344106137629166, 4.62521567510157909213218056545, 5.05060771475604448887299868515, 6.91477314154553955776444743627, 7.44941112397586995703020162067, 8.076779297036741350203023264678, 8.854602173959514193751469262367, 9.751110533370257448011266870498

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