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Properties

Label 2-28e2-28.19-c1-0-10
Degree $2$
Conductor $784$
Sign $0.895 + 0.444i$
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  + (0.5 + 0.866i)3-s + (−1.5 − 0.866i)5-s + (1 − 1.73i)9-s + (−1.5 + 0.866i)11-s − 1.73i·15-s + (4.5 − 2.59i)17-s + (3.5 − 6.06i)19-s + (7.5 + 4.33i)23-s + (−1 − 1.73i)25-s + 5·27-s − 6·29-s + (2.5 + 4.33i)31-s + (−1.5 − 0.866i)33-s + (2.5 − 4.33i)37-s − 6.92i·41-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.670 − 0.387i)5-s + (0.333 − 0.577i)9-s + (−0.452 + 0.261i)11-s − 0.447i·15-s + (1.09 − 0.630i)17-s + (0.802 − 1.39i)19-s + (1.56 + 0.902i)23-s + (−0.200 − 0.346i)25-s + 0.962·27-s − 1.11·29-s + (0.449 + 0.777i)31-s + (−0.261 − 0.150i)33-s + (0.410 − 0.711i)37-s − 1.08i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48938 - 0.348904i\)
\(L(\frac12)\) \(\approx\) \(1.48938 - 0.348904i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (-4.5 + 2.59i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.5 - 4.33i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.5 + 4.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.5 + 2.59i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (1.5 - 0.866i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.5 + 2.59i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-10.5 - 6.06i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−10.10314418106787766696043902162, −9.286453599866327084044658926310, −8.779676994421248773995662252149, −7.47168152636512777600838608495, −7.13292307605143983254463693390, −5.53858770668867169041725447245, −4.78072896388955835417500232974, −3.74122441244136072893837942611, −2.85699053405811446618370403686, −0.879714292953496967311184679090, 1.35165425003764005346321608251, 2.81739195584448249256832099202, 3.73858395510876470245051516241, 4.99524408973515101652134089004, 6.00915689101084006508164803013, 7.14942281939142656670541985136, 7.84213506762761790325756381604, 8.265844554975869043006655133133, 9.605552130623602134850512895814, 10.38668268396802454670584695911

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