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Properties

Label 2-28e2-112.53-c1-0-59
Degree $2$
Conductor $784$
Sign $-0.117 + 0.993i$
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  + (−1.32 − 0.492i)2-s + (0.221 + 0.827i)3-s + (1.51 + 1.30i)4-s + (1.10 − 4.10i)5-s + (0.113 − 1.20i)6-s + (−1.36 − 2.47i)8-s + (1.96 − 1.13i)9-s + (−3.48 + 4.90i)10-s + (1.18 − 0.318i)11-s + (−0.744 + 1.54i)12-s + (1.73 − 1.73i)13-s + 3.64·15-s + (0.592 + 3.95i)16-s + (0.931 − 1.61i)17-s + (−3.15 + 0.535i)18-s + (−3.69 − 0.989i)19-s + ⋯
L(s)  = 1  + (−0.937 − 0.348i)2-s + (0.128 + 0.477i)3-s + (0.757 + 0.652i)4-s + (0.492 − 1.83i)5-s + (0.0463 − 0.492i)6-s + (−0.483 − 0.875i)8-s + (0.653 − 0.377i)9-s + (−1.10 + 1.55i)10-s + (0.358 − 0.0960i)11-s + (−0.214 + 0.445i)12-s + (0.480 − 0.480i)13-s + 0.941·15-s + (0.148 + 0.988i)16-s + (0.226 − 0.391i)17-s + (−0.744 + 0.126i)18-s + (−0.847 − 0.227i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.117 + 0.993i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.117 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.770923 - 0.867083i\)
\(L(\frac12)\) \(\approx\) \(0.770923 - 0.867083i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 + 0.492i)T \)
7 \( 1 \)
good3 \( 1 + (-0.221 - 0.827i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (-1.10 + 4.10i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.18 + 0.318i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.73 + 1.73i)T - 13iT^{2} \)
17 \( 1 + (-0.931 + 1.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.69 + 0.989i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.23 - 0.711i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.181 - 0.181i)T - 29iT^{2} \)
31 \( 1 + (3.23 - 5.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0637 - 0.237i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.440iT - 41T^{2} \)
43 \( 1 + (5.54 + 5.54i)T + 43iT^{2} \)
47 \( 1 + (-3.61 - 6.26i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.69 + 2.59i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.63 - 0.438i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-9.01 - 2.41i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.57 + 9.59i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 11.2iT - 71T^{2} \)
73 \( 1 + (8.13 + 4.69i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.52 - 11.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.06 + 3.06i)T - 83iT^{2} \)
89 \( 1 + (5.66 - 3.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.70T + 97T^{2} \)
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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

−9.929327066904860503636419293174, −9.034001140275457222317189853717, −8.824262118225675222461877292686, −7.86506395663335465534634656796, −6.68086342793459428462924622517, −5.57735217519888100674510358130, −4.50134019696909533128334697460, −3.55809244512462880867455709827, −1.84478877723191776473553447860, −0.811890856901134533985136833791, 1.73046970543535293865609304932, 2.49648411901452915739543848202, 3.91734135715796812979333718283, 5.77033067575024502142470932608, 6.50107000021335626635474525050, 7.04929684584985264984324424609, 7.77223584074638454345864394387, 8.758243442501942466061068023766, 9.940223108349410911593381705752, 10.26874991061864829040395366322

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