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Properties

Label 2-1274-13.10-c1-0-15
Degree $2$
Conductor $1274$
Sign $0.344 - 0.938i$
Analytic cond. $10.1729$
Root an. cond. $3.18950$
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  + (−0.866 + 0.5i)2-s + (0.373 + 0.646i)3-s + (0.499 − 0.866i)4-s + 1.93i·5-s + (−0.646 − 0.373i)6-s + 0.999i·8-s + (1.22 − 2.11i)9-s + (−0.965 − 1.67i)10-s + (3.31 − 1.91i)11-s + 0.746·12-s + (−1.19 + 3.40i)13-s + (−1.24 + 0.721i)15-s + (−0.5 − 0.866i)16-s + (0.129 − 0.224i)17-s + 2.44i·18-s + (2.92 + 1.68i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.215 + 0.373i)3-s + (0.249 − 0.433i)4-s + 0.863i·5-s + (−0.263 − 0.152i)6-s + 0.353i·8-s + (0.407 − 0.705i)9-s + (−0.305 − 0.529i)10-s + (0.999 − 0.577i)11-s + 0.215·12-s + (−0.332 + 0.943i)13-s + (−0.322 + 0.186i)15-s + (−0.125 − 0.216i)16-s + (0.0314 − 0.0545i)17-s + 0.575i·18-s + (0.670 + 0.387i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $0.344 - 0.938i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ 0.344 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.463584500\)
\(L(\frac12)\) \(\approx\) \(1.463584500\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 \)
13 \( 1 + (1.19 - 3.40i)T \)
good3 \( 1 + (-0.373 - 0.646i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 1.93iT - 5T^{2} \)
11 \( 1 + (-3.31 + 1.91i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.129 + 0.224i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.92 - 1.68i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.72 + 4.71i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.31 - 9.21i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.85iT - 31T^{2} \)
37 \( 1 + (-6.27 + 3.62i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.957 + 0.553i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.36iT - 47T^{2} \)
53 \( 1 + 9.04T + 53T^{2} \)
59 \( 1 + (-3.12 - 1.80i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.21 - 3.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.13 + 0.654i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.88 - 3.39i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.97iT - 73T^{2} \)
79 \( 1 + 8.80T + 79T^{2} \)
83 \( 1 - 14.8iT - 83T^{2} \)
89 \( 1 + (-13.4 + 7.75i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.65 + 5.57i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.623484714002351369275683085335, −9.184918858126788406284988540565, −8.324121454098082181833707101443, −7.26429998032932950343014227692, −6.60845567660224071021381826749, −6.06669602889046685707300794914, −4.61585045787107589329122148012, −3.70508870661003778915868888291, −2.66866864818957462451437913779, −1.16332996094048034235618084911, 0.932576451366086151441286609069, 1.89246032549800671603873632341, 3.10090674074066844886436897662, 4.38186663215684989165596710197, 5.14003668420163869047592129307, 6.38995972426277822269151387733, 7.35276614038475023284211724804, 7.990759090763534640892472851728, 8.656278643286915166672209570076, 9.647145193204369263357672355584

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