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Properties

Label 2-28e2-112.93-c1-0-70
Degree $2$
Conductor $784$
Sign $-0.553 - 0.832i$
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  + (−1.29 − 0.560i)2-s + (0.752 − 2.80i)3-s + (1.37 + 1.45i)4-s + (−0.998 − 3.72i)5-s + (−2.55 + 3.22i)6-s + (−0.964 − 2.65i)8-s + (−4.71 − 2.72i)9-s + (−0.792 + 5.39i)10-s + (−1.25 − 0.336i)11-s + (5.11 − 2.75i)12-s + (1.12 + 1.12i)13-s − 11.2·15-s + (−0.238 + 3.99i)16-s + (−0.754 − 1.30i)17-s + (4.59 + 6.17i)18-s + (1.99 − 0.535i)19-s + ⋯
L(s)  = 1  + (−0.918 − 0.396i)2-s + (0.434 − 1.62i)3-s + (0.685 + 0.727i)4-s + (−0.446 − 1.66i)5-s + (−1.04 + 1.31i)6-s + (−0.340 − 0.940i)8-s + (−1.57 − 0.907i)9-s + (−0.250 + 1.70i)10-s + (−0.378 − 0.101i)11-s + (1.47 − 0.795i)12-s + (0.312 + 0.312i)13-s − 2.89·15-s + (−0.0596 + 0.998i)16-s + (−0.182 − 0.316i)17-s + (1.08 + 1.45i)18-s + (0.458 − 0.122i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.553 - 0.832i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (765, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.553 - 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.374773 + 0.699057i\)
\(L(\frac12)\) \(\approx\) \(0.374773 + 0.699057i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 + 0.560i)T \)
7 \( 1 \)
good3 \( 1 + (-0.752 + 2.80i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (0.998 + 3.72i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.25 + 0.336i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.12 - 1.12i)T + 13iT^{2} \)
17 \( 1 + (0.754 + 1.30i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.99 + 0.535i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (4.13 + 2.38i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.10 - 4.10i)T + 29iT^{2} \)
31 \( 1 + (-2.05 - 3.55i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.605 + 2.26i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.45iT - 41T^{2} \)
43 \( 1 + (-5.68 + 5.68i)T - 43iT^{2} \)
47 \( 1 + (-1.79 + 3.11i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.923 - 0.247i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.55 - 0.416i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.39 - 1.17i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (0.558 - 2.08i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 13.8iT - 71T^{2} \)
73 \( 1 + (-12.5 + 7.23i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.889 + 1.54i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.16 - 7.16i)T + 83iT^{2} \)
89 \( 1 + (7.32 + 4.22i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−9.307236636313688364628365675720, −8.668595033654634038821125924311, −8.215104409834354395607474385961, −7.44857629671189308258598704859, −6.64462641859803379634684520433, −5.41049814345173875668371260133, −3.95085605598735964618849884807, −2.52795468428844044125952960527, −1.43836322553286300570258218561, −0.52361456543520067151146935444, 2.53108649149158207639395585745, 3.30872433611074167004636238918, 4.41146193890635703477975829183, 5.74579420755146202984217490354, 6.55118709190969361352466075539, 7.79871302630700158648968570333, 8.162891501933932684030694656527, 9.491362526773835395374555086535, 9.910519936267125975195874943990, 10.66555755439397394873639059542

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