Properties

Label 2-28e2-7.2-c1-0-8
Degree $2$
Conductor $784$
Sign $0.991 + 0.126i$
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 + 1.73i)5-s + (1.5 − 2.59i)9-s + (−2 − 3.46i)11-s + 2·13-s + (3 + 5.19i)17-s + (4 − 6.92i)19-s + (0.500 + 0.866i)25-s + 6·29-s + (4 + 6.92i)31-s + (1 − 1.73i)37-s + 2·41-s + 4·43-s + (3 + 5.19i)45-s + (−4 + 6.92i)47-s + (−3 − 5.19i)53-s + ⋯
L(s)  = 1  + (−0.447 + 0.774i)5-s + (0.5 − 0.866i)9-s + (−0.603 − 1.04i)11-s + 0.554·13-s + (0.727 + 1.26i)17-s + (0.917 − 1.58i)19-s + (0.100 + 0.173i)25-s + 1.11·29-s + (0.718 + 1.24i)31-s + (0.164 − 0.284i)37-s + 0.312·41-s + 0.609·43-s + (0.447 + 0.774i)45-s + (−0.583 + 1.01i)47-s + (−0.412 − 0.713i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50020 - 0.0952033i\)
\(L(\frac12)\) \(\approx\) \(1.50020 - 0.0952033i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4 + 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−10.48294625459876089004200459032, −9.425685092397463660603595718644, −8.530350405450584546386585811456, −7.69809448511971068841847065482, −6.73519707138162428312821304226, −6.07202499312348906035746184998, −4.84894503885296585324137864829, −3.52774099205285981963357840676, −2.99821593164513404316730834034, −0.995768832457415093724212806481, 1.17615431779859805531239590130, 2.62212505602142897624953553630, 4.08748519784175020144851615366, 4.87244005875456822148819751026, 5.67216051759803592901494515599, 7.08398838949866741390065546443, 7.85031518978284808413217539851, 8.354469909226327580269052273759, 9.781253468500353973831116709537, 10.00105542913798980216577191462

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