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Properties

Label 2-28e2-112.109-c1-0-27
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: $\chi_{784} (557, \cdot )$
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

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

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