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Properties

Label 2-28e2-112.53-c1-0-25
Degree $2$
Conductor $784$
Sign $0.963 - 0.266i$
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  + (−1.40 − 0.180i)2-s + (0.152 + 0.569i)3-s + (1.93 + 0.505i)4-s + (−0.414 + 1.54i)5-s + (−0.111 − 0.826i)6-s + (−2.62 − 1.05i)8-s + (2.29 − 1.32i)9-s + (0.859 − 2.09i)10-s + (−5.26 + 1.41i)11-s + (0.00734 + 1.17i)12-s + (4.66 − 4.66i)13-s − 0.943·15-s + (3.48 + 1.95i)16-s + (2.66 − 4.61i)17-s + (−3.46 + 1.44i)18-s + (3.49 + 0.936i)19-s + ⋯
L(s)  = 1  + (−0.991 − 0.127i)2-s + (0.0880 + 0.328i)3-s + (0.967 + 0.252i)4-s + (−0.185 + 0.691i)5-s + (−0.0454 − 0.337i)6-s + (−0.927 − 0.374i)8-s + (0.765 − 0.442i)9-s + (0.271 − 0.662i)10-s + (−1.58 + 0.425i)11-s + (0.00212 + 0.340i)12-s + (1.29 − 1.29i)13-s − 0.243·15-s + (0.872 + 0.489i)16-s + (0.646 − 1.12i)17-s + (−0.815 + 0.340i)18-s + (0.802 + 0.214i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.963 - 0.266i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.963 - 0.266i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04185 + 0.141571i\)
\(L(\frac12)\) \(\approx\) \(1.04185 + 0.141571i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.180i)T \)
7 \( 1 \)
good3 \( 1 + (-0.152 - 0.569i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.414 - 1.54i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (5.26 - 1.41i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-4.66 + 4.66i)T - 13iT^{2} \)
17 \( 1 + (-2.66 + 4.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.49 - 0.936i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.25 + 1.30i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.22 - 1.22i)T - 29iT^{2} \)
31 \( 1 + (-0.416 + 0.722i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.62 - 6.05i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.263iT - 41T^{2} \)
43 \( 1 + (-1.25 - 1.25i)T + 43iT^{2} \)
47 \( 1 + (-5.37 - 9.31i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0650 - 0.0174i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.92 + 1.32i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.09 + 1.63i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-3.48 - 12.9i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 2.05iT - 71T^{2} \)
73 \( 1 + (4.74 + 2.74i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.60 + 4.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.84 + 5.84i)T - 83iT^{2} \)
89 \( 1 + (-5.47 + 3.16i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 18.8T + 97T^{2} \)
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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

−10.35525071079558647247441696680, −9.668087820957661544503888810237, −8.670202569046619017123450716137, −7.63475210631029027687068299588, −7.31612839948196631246349542255, −6.10338804426199998598696969817, −5.05096692302482973890332768010, −3.35869275167312197862196416972, −2.84825792562211675965315216812, −1.00530556942547211105711308304, 1.03439008654300428947823286170, 2.13938276438629344795811400007, 3.65140996178581237238344039672, 5.07259146988896705056436382362, 5.99877405762576488395796115862, 7.08156871483947988583753548723, 7.84415778249933847721525923211, 8.510562349432028317670872940798, 9.205197234035135318179340415508, 10.33826158574437588619646330706

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