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Properties

Label 2-28e2-196.111-c1-0-26
Degree $2$
Conductor $784$
Sign $-0.972 - 0.232i$
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  + (−0.969 − 1.21i)3-s + (−1.39 + 1.10i)5-s + (0.267 − 2.63i)7-s + (0.129 − 0.568i)9-s + (2.43 − 0.556i)11-s + (−5.14 + 1.17i)13-s + (2.69 + 0.615i)15-s + (0.0272 − 0.0565i)17-s + 1.00·19-s + (−3.45 + 2.22i)21-s + (0.107 + 0.222i)23-s + (−0.409 + 1.79i)25-s + (−5.01 + 2.41i)27-s + (−4.86 − 2.34i)29-s − 5.33·31-s + ⋯
L(s)  = 1  + (−0.559 − 0.701i)3-s + (−0.621 + 0.495i)5-s + (0.101 − 0.994i)7-s + (0.0432 − 0.189i)9-s + (0.734 − 0.167i)11-s + (−1.42 + 0.325i)13-s + (0.695 + 0.158i)15-s + (0.00661 − 0.0137i)17-s + 0.229·19-s + (−0.754 + 0.485i)21-s + (0.0223 + 0.0463i)23-s + (−0.0818 + 0.358i)25-s + (−0.965 + 0.465i)27-s + (−0.902 − 0.434i)29-s − 0.958·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.972 - 0.232i$
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: \((2,\ 784,\ (\ :1/2),\ -0.972 - 0.232i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0353955 + 0.300296i\)
\(L(\frac12)\) \(\approx\) \(0.0353955 + 0.300296i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.267 + 2.63i)T \)
good3 \( 1 + (0.969 + 1.21i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (1.39 - 1.10i)T + (1.11 - 4.87i)T^{2} \)
11 \( 1 + (-2.43 + 0.556i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (5.14 - 1.17i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (-0.0272 + 0.0565i)T + (-10.5 - 13.2i)T^{2} \)
19 \( 1 - 1.00T + 19T^{2} \)
23 \( 1 + (-0.107 - 0.222i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (4.86 + 2.34i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + 5.33T + 31T^{2} \)
37 \( 1 + (2.06 + 0.996i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (2.80 - 2.23i)T + (9.12 - 39.9i)T^{2} \)
43 \( 1 + (1.83 + 1.45i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (2.40 + 10.5i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (3.85 - 1.85i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (3.91 - 4.90i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (5.94 - 12.3i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 - 5.43iT - 67T^{2} \)
71 \( 1 + (-1.70 - 3.54i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-5.36 - 1.22i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + 14.8iT - 79T^{2} \)
83 \( 1 + (-0.714 + 3.13i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-7.71 - 1.75i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 - 4.97iT - 97T^{2} \)
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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

−9.901774588448975740054594086027, −9.074193279378720421113579351507, −7.65858604576573883449342893596, −7.22673028908597018178252757568, −6.63465332362995391524640632898, −5.46233530491469300298051998655, −4.23437785642364029887677549619, −3.38106576195641601300165355996, −1.67144779257460826255187816387, −0.15936184595511589319745092504, 2.02965988443109051152190971040, 3.50610294873900450160760838575, 4.73860063831185041937985059136, 5.11412031057289134001915241159, 6.19665441686416432137608431393, 7.43973893677930124865167593393, 8.194854677632985979869746297575, 9.297558623497115004479066491399, 9.727028256469874435224826481691, 10.87741223132812388313676660871

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