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Properties

Label 2-28e2-112.37-c1-0-21
Degree $2$
Conductor $784$
Sign $0.868 + 0.496i$
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  + (−0.366 − 1.36i)2-s + (−1.73 + i)4-s + (−2.73 + 0.732i)5-s + (2 + 1.99i)8-s + (−2.59 − 1.5i)9-s + (2 + 3.46i)10-s + (0.366 − 1.36i)11-s + (1.99 − 3.46i)16-s + (1 + 1.73i)17-s + (−1.09 + 4.09i)18-s + (0.732 + 2.73i)19-s + (3.99 − 4i)20-s − 2·22-s + (5.19 + 3i)23-s + (2.59 − 1.5i)25-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.5i)4-s + (−1.22 + 0.327i)5-s + (0.707 + 0.707i)8-s + (−0.866 − 0.5i)9-s + (0.632 + 1.09i)10-s + (0.110 − 0.411i)11-s + (0.499 − 0.866i)16-s + (0.242 + 0.420i)17-s + (−0.258 + 0.965i)18-s + (0.167 + 0.626i)19-s + (0.894 − 0.894i)20-s − 0.426·22-s + (1.08 + 0.625i)23-s + (0.519 − 0.300i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.868 + 0.496i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.868 + 0.496i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.797449 - 0.211903i\)
\(L(\frac12)\) \(\approx\) \(0.797449 - 0.211903i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
7 \( 1 \)
good3 \( 1 + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (2.73 - 0.732i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.366 + 1.36i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.732 - 2.73i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7 + 7i)T - 29iT^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.83 + 1.83i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 + (1 + i)T + 43iT^{2} \)
47 \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.366 - 1.36i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.92 + 10.9i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.19 - 8.19i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (4.09 + 1.09i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (5.19 - 3i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (10 - 10i)T - 83iT^{2} \)
89 \( 1 + (-12.1 - 7i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−10.35807575818290914975712381836, −9.468586194340885692145812820740, −8.387207096534815025787499038861, −8.109144092927065874584297971438, −6.91616014821181003139031051410, −5.67296716132407692547742911738, −4.41732499685160149464855849702, −3.50256579729986400123455774246, −2.81538966344694079323549728798, −0.898079825164541426494535154774, 0.67864616814371836715876131001, 2.92701211983350709450147268654, 4.36436043551289196253241966855, 4.92067506020894902827078456880, 6.04458061161479931288217647264, 7.14754594518370710204025907616, 7.72750610368812865805904455185, 8.589609497401700871351836564486, 9.104216204459708264661963510658, 10.29366483940835756804359700611

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