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Properties

Label 2-28e2-112.93-c1-0-19
Degree $2$
Conductor $784$
Sign $-0.140 - 0.990i$
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  + (−0.228 + 1.39i)2-s + (−1.89 − 0.637i)4-s + (1.32 − 2.49i)8-s + (2.59 + 1.5i)9-s + (−0.882 − 0.236i)11-s + (3.18 + 2.41i)16-s + (−2.68 + 3.28i)18-s + (0.531 − 1.17i)22-s + (4.58 + 2.64i)23-s + (4.33 − 2.5i)25-s + (4.29 + 4.29i)29-s + (−4.10 + 3.89i)32-s + (−3.96 − 4.5i)36-s + (3.03 + 11.3i)37-s + (−8.64 + 8.64i)43-s + (1.52 + 1.01i)44-s + ⋯
L(s)  = 1  + (−0.161 + 0.986i)2-s + (−0.947 − 0.318i)4-s + (0.467 − 0.883i)8-s + (0.866 + 0.5i)9-s + (−0.265 − 0.0712i)11-s + (0.796 + 0.604i)16-s + (−0.633 + 0.773i)18-s + (0.113 − 0.250i)22-s + (0.955 + 0.551i)23-s + (0.866 − 0.5i)25-s + (0.796 + 0.796i)29-s + (−0.725 + 0.688i)32-s + (−0.661 − 0.750i)36-s + (0.498 + 1.86i)37-s + (−1.31 + 1.31i)43-s + (0.229 + 0.152i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.140 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.140 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.140 - 0.990i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (765, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.140 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.866240 + 0.997496i\)
\(L(\frac12)\) \(\approx\) \(0.866240 + 0.997496i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.228 - 1.39i)T \)
7 \( 1 \)
good3 \( 1 + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.882 + 0.236i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.58 - 2.64i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.29 - 4.29i)T + 29iT^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.03 - 11.3i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (8.64 - 8.64i)T - 43iT^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.398 + 0.106i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-3.63 + 13.5i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 16iT - 71T^{2} \)
73 \( 1 + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.93 + 13.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−10.26434846232634279520032504509, −9.644478848196246216740512976022, −8.613903616542197989738837344133, −7.933137874546674686205678407124, −7.01693054205457739105550117192, −6.38084816845729344329305730966, −5.07365342656790343622747624138, −4.60300778349336930556941440293, −3.16845378797582867861243399937, −1.28399152240201202595641883587, 0.861290944118266461542252124181, 2.26855580803500940522429810017, 3.44922130831009214374022226393, 4.40629344518459641706403713553, 5.31560176746251413749418083462, 6.69078940225150487799742157829, 7.59667507334742621823712648903, 8.651763197115800951417871578794, 9.333003080063832611572950451320, 10.18193812475360880991153129308

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