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Properties

Label 2-31e2-31.25-c1-0-50
Degree $2$
Conductor $961$
Sign $-0.778 + 0.628i$
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 + (0.5 + 0.866i)3-s − 1.61·4-s + (1.30 − 2.26i)5-s + (−0.309 − 0.535i)6-s + (−1.5 − 2.59i)7-s + 2.23·8-s + (1 − 1.73i)9-s + (−0.809 + 1.40i)10-s + (0.381 − 0.661i)11-s + (−0.809 − 1.40i)12-s + (−2.42 + 4.20i)13-s + (0.927 + 1.60i)14-s + 2.61·15-s + 1.85·16-s + (−0.118 − 0.204i)17-s + ⋯
L(s)  = 1  − 0.437·2-s + (0.288 + 0.499i)3-s − 0.809·4-s + (0.585 − 1.01i)5-s + (−0.126 − 0.218i)6-s + (−0.566 − 0.981i)7-s + 0.790·8-s + (0.333 − 0.577i)9-s + (−0.255 + 0.443i)10-s + (0.115 − 0.199i)11-s + (−0.233 − 0.404i)12-s + (−0.673 + 1.16i)13-s + (0.247 + 0.429i)14-s + 0.675·15-s + 0.463·16-s + (−0.0286 − 0.0495i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 + 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.778 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.778 + 0.628i$
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: \((2,\ 961,\ (\ :1/2),\ -0.778 + 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.212822 - 0.602545i\)
\(L(\frac12)\) \(\approx\) \(0.212822 - 0.602545i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.30 + 2.26i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.381 + 0.661i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.42 - 4.20i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.118 + 0.204i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.47T + 23T^{2} \)
29 \( 1 + 8.61T + 29T^{2} \)
37 \( 1 + (0.118 + 0.204i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.23 - 5.60i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.30 - 3.99i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.38T + 47T^{2} \)
53 \( 1 + (-6.35 + 11.0i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.73 + 8.20i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 6.94T + 61T^{2} \)
67 \( 1 + (-2.11 + 3.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.0450 - 0.0780i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.28 + 7.41i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.04 + 3.54i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.38T + 89T^{2} \)
97 \( 1 + 5.29T + 97T^{2} \)
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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

−9.521105813253484560612593464035, −9.208411918233403504768112107082, −8.341352074780228784492968584256, −7.24945694673166805760394367961, −6.34976605596494336238259647734, −5.00641357238536173523840645249, −4.35100566033226139829000256197, −3.62664573896987501163419635451, −1.71190288189301564946508895301, −0.33299250415739078388170106211, 1.87814850913932960187901449661, 2.73666079006984398451906007212, 3.99208839478885984526288914227, 5.40772206936483331245347303077, 6.00782418633550723456298887717, 7.21452330103488761703395776010, 7.85051196984569619879995924242, 8.686935751897333952221350564520, 9.603877753570966576602126385689, 10.23869091155144907052134976126

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