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Properties

Label 2-31e2-31.25-c1-0-43
Degree $2$
Conductor $961$
Sign $0.998 + 0.0543i$
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  + 2.61·2-s + (−0.437 − 0.756i)3-s + 4.85·4-s + (−1.11 + 1.93i)5-s + (−1.14 − 1.98i)6-s + (0.5 + 0.866i)7-s + 7.47·8-s + (1.11 − 1.93i)9-s + (−2.92 + 5.06i)10-s + (2.12 − 3.67i)11-s + (−2.12 − 3.67i)12-s + (−1.31 + 2.27i)13-s + (1.30 + 2.26i)14-s + 1.95·15-s + 9.85·16-s + (1.85 + 3.20i)17-s + ⋯
L(s)  = 1  + 1.85·2-s + (−0.252 − 0.437i)3-s + 2.42·4-s + (−0.499 + 0.866i)5-s + (−0.467 − 0.809i)6-s + (0.188 + 0.327i)7-s + 2.64·8-s + (0.372 − 0.645i)9-s + (−0.925 + 1.60i)10-s + (0.639 − 1.10i)11-s + (−0.612 − 1.06i)12-s + (−0.363 + 0.629i)13-s + (0.349 + 0.605i)14-s + 0.504·15-s + 2.46·16-s + (0.448 + 0.777i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.998 + 0.0543i$
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: \((2,\ 961,\ (\ :1/2),\ 0.998 + 0.0543i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.46954 - 0.121620i\)
\(L(\frac12)\) \(\approx\) \(4.46954 - 0.121620i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.61T + 2T^{2} \)
3 \( 1 + (0.437 + 0.756i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.11 - 1.93i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.12 + 3.67i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.31 - 2.27i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.85 - 3.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.62T + 23T^{2} \)
29 \( 1 - 0.540T + 29T^{2} \)
37 \( 1 + (-2.12 - 3.67i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.736 + 1.27i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.84 + 8.39i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.70T + 47T^{2} \)
53 \( 1 + (6.86 - 11.8i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.97 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 13.9T + 61T^{2} \)
67 \( 1 + (3 - 5.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.736 + 1.27i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.12 + 3.67i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.810 + 1.40i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.58 - 2.73i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−10.57834116317414870566012215349, −9.199276997336417490163307969275, −7.911207374149052494496779286828, −6.94615173977708345433276483146, −6.47615639543392476100370779117, −5.78306069819590006298183690100, −4.64360176785778363453040903576, −3.63720423363257536236207110281, −3.09020429810435355713124070890, −1.65059763698703105866690815670, 1.59923253466223188326047614869, 3.01279340836675752018292135730, 4.18573898230919691209262755907, 4.73580179569913827561543598410, 5.14902909776941271178881787967, 6.34503658844680767829182354687, 7.36666265665117781258647252896, 7.903965346481677371168237932701, 9.438536324832988088236414253496, 10.30650436876473413235527298366

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