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Properties

Label 2-28e2-112.109-c1-0-12
Degree $2$
Conductor $784$
Sign $0.342 - 0.939i$
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  + (−0.246 − 1.39i)2-s + (−2.51 + 0.672i)3-s + (−1.87 + 0.687i)4-s + (2.91 + 0.780i)5-s + (1.55 + 3.33i)6-s + (1.42 + 2.44i)8-s + (3.25 − 1.87i)9-s + (0.367 − 4.24i)10-s + (0.838 + 3.12i)11-s + (4.25 − 2.98i)12-s + (−2.52 − 2.52i)13-s − 7.83·15-s + (3.05 − 2.58i)16-s + (−0.201 + 0.348i)17-s + (−3.41 − 4.06i)18-s + (0.373 − 1.39i)19-s + ⋯
L(s)  = 1  + (−0.174 − 0.984i)2-s + (−1.44 + 0.388i)3-s + (−0.939 + 0.343i)4-s + (1.30 + 0.348i)5-s + (0.635 + 1.35i)6-s + (0.502 + 0.864i)8-s + (1.08 − 0.626i)9-s + (0.116 − 1.34i)10-s + (0.252 + 0.943i)11-s + (1.22 − 0.862i)12-s + (−0.699 − 0.699i)13-s − 2.02·15-s + (0.763 − 0.645i)16-s + (−0.0487 + 0.0844i)17-s + (−0.805 − 0.958i)18-s + (0.0855 − 0.319i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.497941 + 0.348438i\)
\(L(\frac12)\) \(\approx\) \(0.497941 + 0.348438i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.246 + 1.39i)T \)
7 \( 1 \)
good3 \( 1 + (2.51 - 0.672i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-2.91 - 0.780i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.838 - 3.12i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.52 + 2.52i)T + 13iT^{2} \)
17 \( 1 + (0.201 - 0.348i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.373 + 1.39i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (7.89 - 4.55i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.47 - 1.47i)T + 29iT^{2} \)
31 \( 1 + (2.12 - 3.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.94 - 0.520i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 8.96iT - 41T^{2} \)
43 \( 1 + (0.997 - 0.997i)T - 43iT^{2} \)
47 \( 1 + (-2.09 - 3.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.488 - 1.82i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.636 - 2.37i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.685 + 2.55i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (11.6 - 3.12i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 0.451iT - 71T^{2} \)
73 \( 1 + (-9.40 - 5.43i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.31 - 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.742 + 0.742i)T + 83iT^{2} \)
89 \( 1 + (11.1 - 6.41i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−10.41709438361165608888564706415, −9.832969267996967761533988871960, −9.469277590372533371870813474256, −7.953040120348557466913317243680, −6.76045471812708583508781518130, −5.75454934414000398259274407119, −5.14350822972900050461236086163, −4.20064822613765812683550765683, −2.66293091700693903245001026236, −1.47192998756165692319458525802, 0.40429563985924580823003544241, 1.86876968316044570975703309622, 4.20767598891789797501792524512, 5.23719811640062732736816860780, 5.91650716442422345516091883315, 6.31108538710396725297928682638, 7.22201909952110057824361011047, 8.378670364875690762282198227710, 9.312252209353861798785972813826, 10.04786301309682036346098508679

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