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Properties

Label 2-28e2-196.111-c1-0-22
Degree $2$
Conductor $784$
Sign $0.498 + 0.866i$
Analytic cond. $6.26027$
Root an. cond. $2.50205$
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  + (0.560 + 0.702i)3-s + (2.19 − 1.74i)5-s + (−2.31 − 1.28i)7-s + (0.487 − 2.13i)9-s + (0.0549 − 0.0125i)11-s + (0.507 − 0.115i)13-s + (2.45 + 0.560i)15-s + (1.01 − 2.11i)17-s − 0.578·19-s + (−0.390 − 2.34i)21-s + (−2.90 − 6.02i)23-s + (0.634 − 2.78i)25-s + (4.20 − 2.02i)27-s + (−1.97 − 0.949i)29-s − 0.487·31-s + ⋯
L(s)  = 1  + (0.323 + 0.405i)3-s + (0.979 − 0.781i)5-s + (−0.873 − 0.486i)7-s + (0.162 − 0.712i)9-s + (0.0165 − 0.00378i)11-s + (0.140 − 0.0321i)13-s + (0.634 + 0.144i)15-s + (0.247 − 0.513i)17-s − 0.132·19-s + (−0.0851 − 0.512i)21-s + (−0.605 − 1.25i)23-s + (0.126 − 0.556i)25-s + (0.809 − 0.389i)27-s + (−0.366 − 0.176i)29-s − 0.0875·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.498 + 0.866i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.498 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53454 - 0.887678i\)
\(L(\frac12)\) \(\approx\) \(1.53454 - 0.887678i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (2.31 + 1.28i)T \)
good3 \( 1 + (-0.560 - 0.702i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (-2.19 + 1.74i)T + (1.11 - 4.87i)T^{2} \)
11 \( 1 + (-0.0549 + 0.0125i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (-0.507 + 0.115i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (-1.01 + 2.11i)T + (-10.5 - 13.2i)T^{2} \)
19 \( 1 + 0.578T + 19T^{2} \)
23 \( 1 + (2.90 + 6.02i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (1.97 + 0.949i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + 0.487T + 31T^{2} \)
37 \( 1 + (1.98 + 0.958i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (-2.98 + 2.38i)T + (9.12 - 39.9i)T^{2} \)
43 \( 1 + (-9.28 - 7.40i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (-0.708 - 3.10i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-7.43 + 3.57i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (5.54 - 6.95i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-0.642 + 1.33i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 - 1.50iT - 67T^{2} \)
71 \( 1 + (-2.77 - 5.76i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-2.30 - 0.526i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + 9.91iT - 79T^{2} \)
83 \( 1 + (-2.84 + 12.4i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-5.82 - 1.32i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 - 17.7iT - 97T^{2} \)
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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

−9.992227453064345918739168716260, −9.290747804344598975355397866827, −8.851603918445886168305730617924, −7.58438458951359415464675509295, −6.47472727326245263259146451801, −5.83455903128892626224550620438, −4.61953695289598801174973763278, −3.72259210805133030718187529161, −2.51185657284930100244942839235, −0.876743110472119352396904088643, 1.83280248776196262480269829298, 2.65960035592077802283686340675, 3.76331872883819476630188547171, 5.40376435086819040137741412975, 6.06683737194226210468160294197, 6.93633575734733748664459418165, 7.76007415708837832853497755924, 8.830239485095734943989172718172, 9.666780739084766681835736059116, 10.32277871438050474659097524190

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