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Properties

Label 2-1264-316.135-c1-0-11
Degree $2$
Conductor $1264$
Sign $0.549 - 0.835i$
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  + (0.143 + 0.248i)3-s + (−1.31 + 2.26i)5-s + (0.522 − 0.904i)7-s + (1.45 − 2.52i)9-s + (3.34 + 1.92i)11-s + (0.0743 + 0.128i)13-s − 0.751·15-s + 2.07i·17-s + (3.33 + 1.92i)19-s + 0.299·21-s + (−5.48 − 3.16i)23-s + (−0.934 − 1.61i)25-s + 1.69·27-s + (−4.48 − 2.58i)29-s + (7.96 + 4.60i)31-s + ⋯
L(s)  = 1  + (0.0828 + 0.143i)3-s + (−0.586 + 1.01i)5-s + (0.197 − 0.341i)7-s + (0.486 − 0.842i)9-s + (1.00 + 0.581i)11-s + (0.0206 + 0.0357i)13-s − 0.194·15-s + 0.502i·17-s + (0.764 + 0.441i)19-s + 0.0653·21-s + (−1.14 − 0.659i)23-s + (−0.186 − 0.323i)25-s + 0.326·27-s + (−0.832 − 0.480i)29-s + (1.43 + 0.826i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1264\)    =    \(2^{4} \cdot 79\)
Sign: $0.549 - 0.835i$
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: \((2,\ 1264,\ (\ :1/2),\ 0.549 - 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.672352999\)
\(L(\frac12)\) \(\approx\) \(1.672352999\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.46 - 7.00i)T \)
good3 \( 1 + (-0.143 - 0.248i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.31 - 2.26i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.522 + 0.904i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.34 - 1.92i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.0743 - 0.128i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.07iT - 17T^{2} \)
19 \( 1 + (-3.33 - 1.92i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.48 + 3.16i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.48 + 2.58i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.96 - 4.60i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.737 + 0.425i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.0iT - 41T^{2} \)
43 \( 1 + (-5.84 - 10.1i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.36 - 2.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.26 - 3.03i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.30 - 9.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 6.39iT - 61T^{2} \)
67 \( 1 - 11.6iT - 67T^{2} \)
71 \( 1 + 3.93T + 71T^{2} \)
73 \( 1 + (8.04 - 13.9i)T + (-36.5 - 63.2i)T^{2} \)
83 \( 1 + (-5.05 - 2.92i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 0.247T + 89T^{2} \)
97 \( 1 + 8.68T + 97T^{2} \)
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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

−9.920512494834187685784012847864, −9.105886193301666833971077945780, −8.068117653884106357878633379299, −7.24333856020586414654317199737, −6.68211476743341117967514973743, −5.82585751057967423131043205017, −4.16481366841148520238912592128, −3.97985999285366509649881545278, −2.76457744678064788135817431548, −1.25722321196371809799503625415, 0.839180126806782483653270516296, 2.05921826465882688658883208987, 3.52139900463447582169212783133, 4.46871847631900437049851951195, 5.19352611270652788038337559288, 6.16637581512248028031516569737, 7.31783791946697388315932953221, 7.979716633750116039320737252388, 8.701249270270489714752197383742, 9.401381134096073337401033244250

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