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Properties

Label 2-31e2-31.5-c1-0-33
Degree $2$
Conductor $961$
Sign $0.430 + 0.902i$
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  − 2.69·2-s + (0.709 − 1.22i)3-s + 5.23·4-s + (−0.304 − 0.526i)5-s + (−1.90 + 3.30i)6-s + (0.863 − 1.49i)7-s − 8.70·8-s + (0.492 + 0.853i)9-s + (0.818 + 1.41i)10-s + (−0.668 − 1.15i)11-s + (3.71 − 6.43i)12-s + (1.83 + 3.17i)13-s + (−2.32 + 4.02i)14-s − 0.863·15-s + 12.9·16-s + (−1.38 + 2.39i)17-s + ⋯
L(s)  = 1  − 1.90·2-s + (0.409 − 0.709i)3-s + 2.61·4-s + (−0.136 − 0.235i)5-s + (−0.779 + 1.34i)6-s + (0.326 − 0.565i)7-s − 3.07·8-s + (0.164 + 0.284i)9-s + (0.258 + 0.448i)10-s + (−0.201 − 0.349i)11-s + (1.07 − 1.85i)12-s + (0.509 + 0.881i)13-s + (−0.620 + 1.07i)14-s − 0.222·15-s + 3.23·16-s + (−0.335 + 0.580i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.430 + 0.902i$
Analytic conductor: \(7.67362\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :1/2),\ 0.430 + 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.692586 - 0.437148i\)
\(L(\frac12)\) \(\approx\) \(0.692586 - 0.437148i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.69T + 2T^{2} \)
3 \( 1 + (-0.709 + 1.22i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.304 + 0.526i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.863 + 1.49i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.668 + 1.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.83 - 3.17i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.38 - 2.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.28 + 2.22i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.539T + 23T^{2} \)
29 \( 1 - 8.13T + 29T^{2} \)
37 \( 1 + (-3.87 + 6.70i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.0520 - 0.0901i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.50 - 2.60i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.72T + 47T^{2} \)
53 \( 1 + (-1.39 - 2.42i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.233 - 0.403i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 5.11T + 61T^{2} \)
67 \( 1 + (4.14 + 7.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.37 + 4.12i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.75 - 6.50i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.84 - 8.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.34 + 14.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 + 1.27T + 97T^{2} \)
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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

−9.754790179608485122591711562212, −8.734267704086375852570281135413, −8.378391985405562309952420814242, −7.59787683404715692611277491228, −6.90583066293098230558801085070, −6.19242764625998943852358600930, −4.49415801025223815720873163514, −2.84941541668675047223715592198, −1.80430299730196111070196008227, −0.851275358399873211164007252412, 1.10457259585921827364760772260, 2.56500645966624268305139933430, 3.39180795650378480737942765989, 5.03569879264286407185389638362, 6.28704694125870129728469662212, 7.09832329999099975783495248443, 8.057369577985236044929429643491, 8.608384051746069202932652914833, 9.314695556763106468590291979098, 10.09594238043935084287243562306

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