Properties

Label 2-28e2-112.53-c1-0-19
Degree $2$
Conductor $784$
Sign $0.970 + 0.241i$
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  + (−1.41 + 0.00789i)2-s + (−0.814 − 3.04i)3-s + (1.99 − 0.0223i)4-s + (−0.501 + 1.87i)5-s + (1.17 + 4.29i)6-s + (−2.82 + 0.0473i)8-s + (−5.98 + 3.45i)9-s + (0.694 − 2.65i)10-s + (−1.11 + 0.299i)11-s + (−1.69 − 6.06i)12-s + (−0.00680 + 0.00680i)13-s + 6.10·15-s + (3.99 − 0.0893i)16-s + (−1.52 + 2.63i)17-s + (8.43 − 4.93i)18-s + (4.64 + 1.24i)19-s + ⋯
L(s)  = 1  + (−0.999 + 0.00558i)2-s + (−0.470 − 1.75i)3-s + (0.999 − 0.0111i)4-s + (−0.224 + 0.837i)5-s + (0.480 + 1.75i)6-s + (−0.999 + 0.0167i)8-s + (−1.99 + 1.15i)9-s + (0.219 − 0.838i)10-s + (−0.337 + 0.0904i)11-s + (−0.490 − 1.75i)12-s + (−0.00188 + 0.00188i)13-s + 1.57·15-s + (0.999 − 0.0223i)16-s + (−0.369 + 0.639i)17-s + (1.98 − 1.16i)18-s + (1.06 + 0.285i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.970 + 0.241i$
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: \((2,\ 784,\ (\ :1/2),\ 0.970 + 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.653849 - 0.0800382i\)
\(L(\frac12)\) \(\approx\) \(0.653849 - 0.0800382i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.00789i)T \)
7 \( 1 \)
good3 \( 1 + (0.814 + 3.04i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.501 - 1.87i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.11 - 0.299i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.00680 - 0.00680i)T - 13iT^{2} \)
17 \( 1 + (1.52 - 2.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.64 - 1.24i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.27 + 2.46i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.45 - 1.45i)T - 29iT^{2} \)
31 \( 1 + (2.60 - 4.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.28 + 8.51i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 6.02iT - 41T^{2} \)
43 \( 1 + (-7.17 - 7.17i)T + 43iT^{2} \)
47 \( 1 + (0.796 + 1.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.787 - 0.211i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-7.25 + 1.94i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-9.21 - 2.46i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.70 + 6.35i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 6.77iT - 71T^{2} \)
73 \( 1 + (-3.43 - 1.98i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.81 + 4.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.3 + 11.3i)T - 83iT^{2} \)
89 \( 1 + (3.59 - 2.07i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.390T + 97T^{2} \)
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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

−10.54705036602026550735928101600, −9.252618571162503277078048790897, −8.279635407680457863559983760315, −7.52336114441572342520183104864, −7.02499131636593619418433641348, −6.32038763543909045383542574304, −5.42109694383159113769234000674, −3.16366520225942779336208177855, −2.21653212126219056411726661930, −1.00754451240977035846378067144, 0.61904611961803666188605409858, 2.81026766342537024437343787935, 3.93548906957451786091293795048, 5.07501766311175815323748076649, 5.61375750805670047702535822078, 6.96988972803909343916448373526, 8.111874853796240924782047979471, 9.024098072838259878136281866418, 9.411615770837221862824888043331, 10.15864082551664898677572784788

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