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Properties

Label 2-31e2-31.25-c1-0-51
Degree $2$
Conductor $961$
Sign $0.602 + 0.798i$
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  + 1.61·2-s + (0.5 + 0.866i)3-s + 0.618·4-s + (0.190 − 0.330i)5-s + (0.809 + 1.40i)6-s + (−1.5 − 2.59i)7-s − 2.23·8-s + (1 − 1.73i)9-s + (0.309 − 0.535i)10-s + (2.61 − 4.53i)11-s + (0.309 + 0.535i)12-s + (0.927 − 1.60i)13-s + (−2.42 − 4.20i)14-s + 0.381·15-s − 4.85·16-s + (2.11 + 3.66i)17-s + ⋯
L(s)  = 1  + 1.14·2-s + (0.288 + 0.499i)3-s + 0.309·4-s + (0.0854 − 0.147i)5-s + (0.330 + 0.572i)6-s + (−0.566 − 0.981i)7-s − 0.790·8-s + (0.333 − 0.577i)9-s + (0.0977 − 0.169i)10-s + (0.789 − 1.36i)11-s + (0.0892 + 0.154i)12-s + (0.257 − 0.445i)13-s + (−0.648 − 1.12i)14-s + 0.0986·15-s − 1.21·16-s + (0.513 + 0.889i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 + 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.602 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.602 + 0.798i$
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: \((2,\ 961,\ (\ :1/2),\ 0.602 + 0.798i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.41288 - 1.20148i\)
\(L(\frac12)\) \(\approx\) \(2.41288 - 1.20148i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.61T + 2T^{2} \)
3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.190 + 0.330i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.61 + 4.53i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.927 + 1.60i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.11 - 3.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.47T + 23T^{2} \)
29 \( 1 + 6.38T + 29T^{2} \)
37 \( 1 + (-2.11 - 3.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.23 + 2.14i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.19 - 2.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.61T + 47T^{2} \)
53 \( 1 + (0.354 - 0.613i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.263 + 0.457i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 + (0.118 - 0.204i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.54 + 9.60i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.78 - 10.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.54 - 6.14i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.61T + 89T^{2} \)
97 \( 1 + 18.7T + 97T^{2} \)
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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

−9.848490707728626300813043494726, −9.102145627017218828752787007067, −8.457605032384947317844090793052, −6.99912208761046916675679942152, −6.32439091661178827062059344935, −5.46084277892701151564483091195, −4.32867812154220735062944110075, −3.61139875323094094272934586296, −3.16685279776922432689555603299, −0.881689535144973294210382373815, 1.90235990905596620668966263654, 2.79027063160270637161527285013, 3.97575020830683024461506652017, 4.82655985246766272231881867492, 5.76656416609085664049647129978, 6.63120922657954991949760348294, 7.31774072118768292731251352037, 8.534998898850140214862758480171, 9.341993183077330501068625524682, 9.978219081642395079502770143405

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