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Properties

Label 2-28e2-112.109-c1-0-13
Degree $2$
Conductor $784$
Sign $-0.465 - 0.885i$
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  + (−1.38 − 0.265i)2-s + (−0.331 + 0.0888i)3-s + (1.85 + 0.738i)4-s + (2.28 + 0.613i)5-s + (0.484 − 0.0352i)6-s + (−2.38 − 1.52i)8-s + (−2.49 + 1.44i)9-s + (−3.01 − 1.46i)10-s + (0.684 + 2.55i)11-s + (−0.682 − 0.0798i)12-s + (−2.24 − 2.24i)13-s − 0.813·15-s + (2.90 + 2.74i)16-s + (−2.63 + 4.56i)17-s + (3.85 − 1.33i)18-s + (−1.90 + 7.09i)19-s + ⋯
L(s)  = 1  + (−0.982 − 0.188i)2-s + (−0.191 + 0.0513i)3-s + (0.929 + 0.369i)4-s + (1.02 + 0.274i)5-s + (0.197 − 0.0143i)6-s + (−0.843 − 0.537i)8-s + (−0.831 + 0.480i)9-s + (−0.953 − 0.461i)10-s + (0.206 + 0.770i)11-s + (−0.196 − 0.0230i)12-s + (−0.623 − 0.623i)13-s − 0.210·15-s + (0.727 + 0.686i)16-s + (−0.638 + 1.10i)17-s + (0.907 − 0.315i)18-s + (−0.436 + 1.62i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.465 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.465 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.465 - 0.885i$
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: \((2,\ 784,\ (\ :1/2),\ -0.465 - 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.324945 + 0.537879i\)
\(L(\frac12)\) \(\approx\) \(0.324945 + 0.537879i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 0.265i)T \)
7 \( 1 \)
good3 \( 1 + (0.331 - 0.0888i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-2.28 - 0.613i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.684 - 2.55i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.24 + 2.24i)T + 13iT^{2} \)
17 \( 1 + (2.63 - 4.56i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.90 - 7.09i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.0792 - 0.0457i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.55 + 6.55i)T + 29iT^{2} \)
31 \( 1 + (0.331 - 0.574i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.00 + 0.270i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.43iT - 41T^{2} \)
43 \( 1 + (-5.68 + 5.68i)T - 43iT^{2} \)
47 \( 1 + (-2.65 - 4.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.233 - 0.871i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.48 - 9.27i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.560 - 2.09i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (9.03 - 2.42i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 15.1iT - 71T^{2} \)
73 \( 1 + (4.17 + 2.40i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.53 + 9.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.9 - 10.9i)T + 83iT^{2} \)
89 \( 1 + (1.96 - 1.13i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 17.4T + 97T^{2} \)
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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

−10.31402238896347208564262318118, −9.943160842816456784425050284022, −8.963074921257314165653293742925, −8.095148801750242989235438468088, −7.31441104578554552243307036787, −6.05681754590383106989559798369, −5.74740149537754084827595458929, −4.04070636911266516302141869022, −2.52934751105312663488359080596, −1.81726513131792103877946170860, 0.40538023225519301984458247217, 2.01031952046114395926513308368, 3.02698634642332731692472160384, 4.94645027504479247674477305184, 5.79202979137255992140073142254, 6.61193609699489183261885196748, 7.32742445087888889711885534000, 8.761810400640542627377373897171, 9.096700418353824863823852221656, 9.671638936535601509018223081204

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