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Properties

Label 2-28e2-196.111-c1-0-13
Degree $2$
Conductor $784$
Sign $0.911 - 0.411i$
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  + (0.683 + 0.857i)3-s + (−1.48 + 1.18i)5-s + (2.57 − 0.598i)7-s + (0.400 − 1.75i)9-s + (3.19 − 0.729i)11-s + (4.87 − 1.11i)13-s + (−2.02 − 0.462i)15-s + (−0.421 + 0.874i)17-s − 7.47·19-s + (2.27 + 1.79i)21-s + (−1.73 − 3.61i)23-s + (−0.311 + 1.36i)25-s + (4.73 − 2.28i)27-s + (9.12 + 4.39i)29-s + 0.116·31-s + ⋯
L(s)  = 1  + (0.394 + 0.494i)3-s + (−0.663 + 0.528i)5-s + (0.974 − 0.226i)7-s + (0.133 − 0.584i)9-s + (0.963 − 0.219i)11-s + (1.35 − 0.308i)13-s + (−0.523 − 0.119i)15-s + (−0.102 + 0.212i)17-s − 1.71·19-s + (0.496 + 0.392i)21-s + (−0.362 − 0.752i)23-s + (−0.0623 + 0.273i)25-s + (0.912 − 0.439i)27-s + (1.69 + 0.815i)29-s + 0.0209·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.911 - 0.411i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.911 - 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85989 + 0.400630i\)
\(L(\frac12)\) \(\approx\) \(1.85989 + 0.400630i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2.57 + 0.598i)T \)
good3 \( 1 + (-0.683 - 0.857i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (1.48 - 1.18i)T + (1.11 - 4.87i)T^{2} \)
11 \( 1 + (-3.19 + 0.729i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (-4.87 + 1.11i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (0.421 - 0.874i)T + (-10.5 - 13.2i)T^{2} \)
19 \( 1 + 7.47T + 19T^{2} \)
23 \( 1 + (1.73 + 3.61i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (-9.12 - 4.39i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 - 0.116T + 31T^{2} \)
37 \( 1 + (-8.70 - 4.19i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (5.62 - 4.48i)T + (9.12 - 39.9i)T^{2} \)
43 \( 1 + (-4.92 - 3.92i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (0.387 + 1.69i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (3.77 - 1.81i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (6.73 - 8.44i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-3.39 + 7.03i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + 14.4iT - 67T^{2} \)
71 \( 1 + (3.98 + 8.26i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-1.76 - 0.403i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 - 5.52iT - 79T^{2} \)
83 \( 1 + (3.87 - 16.9i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (8.43 + 1.92i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + 7.90iT - 97T^{2} \)
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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

−10.63211453664542400661306706769, −9.415847727898392975685888901995, −8.405457726581634317001634563981, −8.206893370992328534521514640980, −6.70480011865589503392195870106, −6.23388709979082396278575059033, −4.51284440919232876804352429337, −4.01204172427292274486019734052, −3.01950421329084192276562435367, −1.27611742351612836821307548618, 1.26869405336624025482959026450, 2.30847076776939083481251225692, 4.06911009332456348503920759360, 4.51013231469162271286359391409, 5.90922772172652264535031197495, 6.85379135559768748908077415003, 7.950904570674910153010325634443, 8.412697129355164985601802381679, 9.000091887254064212374540076446, 10.36230392848590140292822214051

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