Properties

Label 2-28e2-112.53-c1-0-48
Degree $2$
Conductor $784$
Sign $0.448 - 0.893i$
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  + (0.114 + 1.40i)2-s + (0.509 + 1.90i)3-s + (−1.97 + 0.323i)4-s + (0.792 − 2.95i)5-s + (−2.62 + 0.935i)6-s + (−0.681 − 2.74i)8-s + (−0.754 + 0.435i)9-s + (4.25 + 0.778i)10-s + (4.22 − 1.13i)11-s + (−1.61 − 3.58i)12-s + (1.75 − 1.75i)13-s + 6.02·15-s + (3.79 − 1.27i)16-s + (2.60 − 4.50i)17-s + (−0.700 − 1.01i)18-s + (1.16 + 0.311i)19-s + ⋯
L(s)  = 1  + (0.0810 + 0.996i)2-s + (0.294 + 1.09i)3-s + (−0.986 + 0.161i)4-s + (0.354 − 1.32i)5-s + (−1.06 + 0.381i)6-s + (−0.240 − 0.970i)8-s + (−0.251 + 0.145i)9-s + (1.34 + 0.246i)10-s + (1.27 − 0.341i)11-s + (−0.467 − 1.03i)12-s + (0.486 − 0.486i)13-s + 1.55·15-s + (0.947 − 0.318i)16-s + (0.631 − 1.09i)17-s + (−0.165 − 0.238i)18-s + (0.266 + 0.0715i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.448 - 0.893i$
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: \((2,\ 784,\ (\ :1/2),\ 0.448 - 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57482 + 0.971687i\)
\(L(\frac12)\) \(\approx\) \(1.57482 + 0.971687i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.114 - 1.40i)T \)
7 \( 1 \)
good3 \( 1 + (-0.509 - 1.90i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.792 + 2.95i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-4.22 + 1.13i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.75 + 1.75i)T - 13iT^{2} \)
17 \( 1 + (-2.60 + 4.50i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.16 - 0.311i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.33 - 3.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.24 - 6.24i)T - 29iT^{2} \)
31 \( 1 + (-1.39 + 2.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.50 + 5.61i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 6.32iT - 41T^{2} \)
43 \( 1 + (-3.05 - 3.05i)T + 43iT^{2} \)
47 \( 1 + (1.80 + 3.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.21 - 1.93i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-9.74 + 2.61i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.42 + 0.380i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.353 + 1.32i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 10.0iT - 71T^{2} \)
73 \( 1 + (-13.1 - 7.58i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.30 - 5.72i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.41 - 7.41i)T - 83iT^{2} \)
89 \( 1 + (2.82 - 1.63i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.66T + 97T^{2} \)
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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

−9.830747978194709805851995565439, −9.445668611641688374710690398814, −8.888297643174294237252119216839, −8.080768883046956497289419576113, −6.97662851859837694303048028581, −5.68390665140495082293670751306, −5.24356896619367654085041093225, −4.13623497660771428653202157875, −3.56689431313586171291193044845, −1.08163997721754771079306208955, 1.45965332224737859390578917047, 2.20803751254678876804620233068, 3.38095817044330353177461722525, 4.29599291779778040019072168339, 6.09386790099862838603956802649, 6.49772203328032817552476947847, 7.63350185855916656235873829175, 8.422328295971261426632945993873, 9.571598420564868972181159692508, 10.17357860903288111092548501940

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