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Properties

Label 2-28e2-112.109-c1-0-29
Degree $2$
Conductor $784$
Sign $0.159 + 0.987i$
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  + (−1.25 − 0.656i)2-s + (−1.95 + 0.523i)3-s + (1.13 + 1.64i)4-s + (−0.959 − 0.256i)5-s + (2.78 + 0.625i)6-s + (−0.348 − 2.80i)8-s + (0.941 − 0.543i)9-s + (1.03 + 0.951i)10-s + (−0.505 − 1.88i)11-s + (−3.08 − 2.61i)12-s + (2.10 + 2.10i)13-s + 2.00·15-s + (−1.40 + 3.74i)16-s + (−2.83 + 4.91i)17-s + (−1.53 + 0.0632i)18-s + (0.165 − 0.616i)19-s + ⋯
L(s)  = 1  + (−0.885 − 0.463i)2-s + (−1.12 + 0.302i)3-s + (0.569 + 0.821i)4-s + (−0.428 − 0.114i)5-s + (1.13 + 0.255i)6-s + (−0.123 − 0.992i)8-s + (0.313 − 0.181i)9-s + (0.326 + 0.300i)10-s + (−0.152 − 0.569i)11-s + (−0.890 − 0.754i)12-s + (0.583 + 0.583i)13-s + 0.518·15-s + (−0.351 + 0.936i)16-s + (−0.688 + 1.19i)17-s + (−0.362 + 0.0149i)18-s + (0.0379 − 0.141i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.159 + 0.987i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.159 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.287217 - 0.244451i\)
\(L(\frac12)\) \(\approx\) \(0.287217 - 0.244451i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.25 + 0.656i)T \)
7 \( 1 \)
good3 \( 1 + (1.95 - 0.523i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (0.959 + 0.256i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.505 + 1.88i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-2.10 - 2.10i)T + 13iT^{2} \)
17 \( 1 + (2.83 - 4.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.165 + 0.616i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (5.92 - 3.42i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.207 + 0.207i)T + 29iT^{2} \)
31 \( 1 + (-3.94 + 6.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.60 - 2.57i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.40iT - 41T^{2} \)
43 \( 1 + (-3.65 + 3.65i)T - 43iT^{2} \)
47 \( 1 + (-0.144 - 0.250i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.04 + 7.62i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.60 + 13.4i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.969 + 3.61i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-10.0 + 2.69i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 11.5iT - 71T^{2} \)
73 \( 1 + (-0.310 - 0.179i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.84 + 6.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.424 - 0.424i)T + 83iT^{2} \)
89 \( 1 + (-15.2 + 8.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−10.19878761623853973364232361739, −9.457762355073792495188129054453, −8.311105719395842233491155130208, −7.88093780119573664792760770584, −6.37445805533469685002371972568, −6.03968463671814746127371185041, −4.47293900215290904357547308404, −3.66860048958708332795281766869, −2.04001605171245358841359527954, −0.38633929440151112165487376897, 0.935813266421926753150845721841, 2.59866831683274430822163218038, 4.42303474783581914908085256826, 5.50703514196591534576044558487, 6.21187171216498457996711793226, 7.04736623084997467708501294055, 7.78283411210433040542623369427, 8.705215513655545171696114903779, 9.695284786738076283670297113931, 10.53617562651883200377241418063

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