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Properties

Label 2-1274-91.51-c1-0-20
Degree $2$
Conductor $1274$
Sign $0.756 - 0.653i$
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 + (−0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (2.59 + 1.5i)5-s − 0.999i·6-s + 0.999i·8-s + (1 − 1.73i)9-s + (1.5 + 2.59i)10-s + (0.499 − 0.866i)12-s + (−2 + 3i)13-s − 3i·15-s + (−0.5 + 0.866i)16-s + (1.5 + 2.59i)17-s + (1.73 − i)18-s + (5.19 + 3i)19-s + 3i·20-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + (1.16 + 0.670i)5-s − 0.408i·6-s + 0.353i·8-s + (0.333 − 0.577i)9-s + (0.474 + 0.821i)10-s + (0.144 − 0.249i)12-s + (−0.554 + 0.832i)13-s − 0.774i·15-s + (−0.125 + 0.216i)16-s + (0.363 + 0.630i)17-s + (0.408 − 0.235i)18-s + (1.19 + 0.688i)19-s + 0.670i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $0.756 - 0.653i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ 0.756 - 0.653i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.744505768\)
\(L(\frac12)\) \(\approx\) \(2.744505768\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 3i)T \)
good3 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.59 - 1.5i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.19 - 3i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.59 - 1.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (2.59 + 1.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.19 - 3i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.3 + 6i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 15iT - 71T^{2} \)
73 \( 1 + (-5.19 + 3i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + (-5.19 - 3i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 12iT - 97T^{2} \)
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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

−9.795017444203280470936003003825, −9.072754156700747130232002672901, −7.81755528338631880810873813177, −7.01534732486635067279065776935, −6.38739812144408266708957296333, −5.83764207149067862193211377068, −4.82880216076096719655633768152, −3.65743169481239629494540953851, −2.56373061474021031544810769533, −1.46931876729662613608675789985, 1.12780256515364888903745408009, 2.34924094196406718654719032890, 3.41108451277968069361787984989, 4.80828917806803805810119054582, 5.20891264970158034807408831411, 5.75794983489438271745860993838, 7.04525810294726909293782920064, 7.86323168614981426475464671924, 9.237195650937738707766655380089, 9.687022319853740857327447430957

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