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Properties

Label 2-31e2-31.25-c1-0-62
Degree $2$
Conductor $961$
Sign $-0.613 + 0.789i$
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  + 2.30·2-s + (−1.27 − 2.20i)3-s + 3.32·4-s + (1.24 − 2.16i)5-s + (−2.93 − 5.08i)6-s + (−0.800 − 1.38i)7-s + 3.05·8-s + (−1.73 + 3.00i)9-s + (2.88 − 4.99i)10-s + (0.366 − 0.635i)11-s + (−4.22 − 7.32i)12-s + (−0.947 + 1.64i)13-s + (−1.84 − 3.20i)14-s − 6.35·15-s + 0.404·16-s + (−2.18 − 3.79i)17-s + ⋯
L(s)  = 1  + 1.63·2-s + (−0.734 − 1.27i)3-s + 1.66·4-s + (0.558 − 0.967i)5-s + (−1.19 − 2.07i)6-s + (−0.302 − 0.524i)7-s + 1.08·8-s + (−0.577 + 1.00i)9-s + (0.911 − 1.57i)10-s + (0.110 − 0.191i)11-s + (−1.22 − 2.11i)12-s + (−0.262 + 0.455i)13-s + (−0.493 − 0.855i)14-s − 1.64·15-s + 0.101·16-s + (−0.531 − 0.919i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.613 + 0.789i$
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: \((2,\ 961,\ (\ :1/2),\ -0.613 + 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37697 - 2.81349i\)
\(L(\frac12)\) \(\approx\) \(1.37697 - 2.81349i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.30T + 2T^{2} \)
3 \( 1 + (1.27 + 2.20i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.24 + 2.16i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.800 + 1.38i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.366 + 0.635i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.947 - 1.64i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.18 + 3.79i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.31 - 4.01i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.11T + 23T^{2} \)
29 \( 1 - 0.128T + 29T^{2} \)
37 \( 1 + (-4.21 - 7.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.68 + 6.38i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.115 - 0.199i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.03T + 47T^{2} \)
53 \( 1 + (-2.86 + 4.96i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.75 + 8.23i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 7.84T + 61T^{2} \)
67 \( 1 + (2.41 - 4.17i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.70 + 2.94i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.34 + 2.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.26 - 3.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.33 + 2.31i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.20T + 89T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−9.813534897138017945954676345135, −8.850376217856527043638476686155, −7.51398743132902673071589270131, −6.81617466597963748892242066384, −6.19551103892229484097646424726, −5.27201301187272309132530559998, −4.77907836488175425323656830864, −3.46648557882021410782280805253, −2.13373649160709631226318186345, −0.950854191093059356925583155111, 2.51498146174395769428039106886, 3.22239000123470119007642853608, 4.29653086198143753681670654578, 5.02278504067194396791979402797, 5.81668488696744446998229772400, 6.37379001013987406989725162338, 7.27284129991896581043769771520, 9.013771348033184295911881443674, 9.735380525097993182501158725239, 10.76434610866722030127578687590

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