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Properties

Label 2-31e2-31.25-c1-0-12
Degree $2$
Conductor $961$
Sign $0.695 + 0.718i$
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  − 2.41·2-s + (−1.20 − 2.09i)3-s + 3.82·4-s + (−0.5 + 0.866i)5-s + (2.91 + 5.04i)6-s + (−1.20 − 2.09i)7-s − 4.41·8-s + (−1.41 + 2.44i)9-s + (1.20 − 2.09i)10-s + (−2.62 + 4.54i)11-s + (−4.62 − 8.00i)12-s + (0.914 − 1.58i)13-s + (2.91 + 5.04i)14-s + 2.41·15-s + 2.99·16-s + (−0.0857 − 0.148i)17-s + ⋯
L(s)  = 1  − 1.70·2-s + (−0.696 − 1.20i)3-s + 1.91·4-s + (−0.223 + 0.387i)5-s + (1.18 + 2.06i)6-s + (−0.456 − 0.790i)7-s − 1.56·8-s + (−0.471 + 0.816i)9-s + (0.381 − 0.661i)10-s + (−0.790 + 1.36i)11-s + (−1.33 − 2.31i)12-s + (0.253 − 0.439i)13-s + (0.778 + 1.34i)14-s + 0.623·15-s + 0.749·16-s + (−0.0208 − 0.0360i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.695 + 0.718i$
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: \((2,\ 961,\ (\ :1/2),\ 0.695 + 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.339712 - 0.144078i\)
\(L(\frac12)\) \(\approx\) \(0.339712 - 0.144078i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.41T + 2T^{2} \)
3 \( 1 + (1.20 + 2.09i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.20 + 2.09i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.62 - 4.54i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.914 + 1.58i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.0857 + 0.148i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.792 + 1.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 1.17T + 29T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.74 - 8.21i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.44 - 7.70i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.65T + 47T^{2} \)
53 \( 1 + (-0.0857 + 0.148i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.03 - 8.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 2.82T + 61T^{2} \)
67 \( 1 + (-2.62 + 4.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.03 + 12.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.91 + 3.31i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.62 + 13.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.03 + 3.52i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.4T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−10.00236395032291648286239441240, −9.128881209155878468568740232605, −7.930020210588010604269846672159, −7.45109860846574557878382553865, −6.91708376148186820785240447454, −6.26711509247463542873291229578, −4.82137948875607305104108147437, −2.98802687226150834318571769662, −1.75889774784381836786252417431, −0.65803658517099809188648204106, 0.59395768128095195631870114214, 2.47352233885939227445554370015, 3.73950660478233899369730087343, 5.11637598512137826302538039205, 5.87233183573968141490327745271, 6.81743388615883238493116759012, 8.087493627485595907327946366819, 8.746731029083545035515768932062, 9.206553312501218602625368785600, 10.15085408833124764211542004153

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