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Properties

Label 2-1264-316.135-c1-0-23
Degree $2$
Conductor $1264$
Sign $0.969 - 0.245i$
Analytic cond. $10.0930$
Root an. cond. $3.17696$
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  + (0.799 + 1.38i)3-s + (1.24 − 2.15i)5-s + (−0.627 + 1.08i)7-s + (0.220 − 0.382i)9-s + (4.25 + 2.45i)11-s + (0.756 + 1.31i)13-s + 3.97·15-s − 5.86i·17-s + (−4.82 − 2.78i)19-s − 2.00·21-s + (2.76 + 1.59i)23-s + (−0.595 − 1.03i)25-s + 5.50·27-s + (3.75 + 2.16i)29-s + (2.49 + 1.44i)31-s + ⋯
L(s)  = 1  + (0.461 + 0.799i)3-s + (0.556 − 0.963i)5-s + (−0.237 + 0.411i)7-s + (0.0735 − 0.127i)9-s + (1.28 + 0.740i)11-s + (0.209 + 0.363i)13-s + 1.02·15-s − 1.42i·17-s + (−1.10 − 0.639i)19-s − 0.438·21-s + (0.576 + 0.333i)23-s + (−0.119 − 0.206i)25-s + 1.05·27-s + (0.697 + 0.402i)29-s + (0.448 + 0.259i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1264\)    =    \(2^{4} \cdot 79\)
Sign: $0.969 - 0.245i$
Analytic conductor: \(10.0930\)
Root analytic conductor: \(3.17696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1264} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1264,\ (\ :1/2),\ 0.969 - 0.245i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.319842296\)
\(L(\frac12)\) \(\approx\) \(2.319842296\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
79 \( 1 + (0.689 - 8.86i)T \)
good3 \( 1 + (-0.799 - 1.38i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.24 + 2.15i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.627 - 1.08i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.25 - 2.45i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.756 - 1.31i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.86iT - 17T^{2} \)
19 \( 1 + (4.82 + 2.78i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.76 - 1.59i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.75 - 2.16i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.49 - 1.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.96 - 2.86i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 + (1.55 + 2.69i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.56 - 2.70i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.99 - 4.61i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.13 + 3.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 4.46iT - 61T^{2} \)
67 \( 1 + 7.16iT - 67T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + (-3.18 + 5.51i)T + (-36.5 - 63.2i)T^{2} \)
83 \( 1 + (-0.643 - 0.371i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−9.420467381521333013908691889783, −9.050745792385851694564573379813, −8.648635428632088739279062127130, −7.06766761580575228325185655005, −6.50926549629684442520620089247, −5.20932740774999948880016896770, −4.60557365025500526630060971023, −3.74140128286129801475428897428, −2.51335908129148880576846222937, −1.19128768286057577519947611116, 1.26613659647184922879854000869, 2.30169087058069406372757986250, 3.36009557825065431399345003833, 4.25309688427622918830611297337, 5.85913656541236004217010073651, 6.56984884158170485903905013634, 6.89744356440563420551812989554, 8.280820674883922884581428850541, 8.439224223255716774800936577165, 9.790946748209802341579071054130

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