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Properties

Label 2-28e2-112.19-c1-0-26
Degree $2$
Conductor $784$
Sign $0.643 - 0.765i$
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  + (−0.830 − 1.14i)2-s + (−0.472 + 1.76i)3-s + (−0.621 + 1.90i)4-s + (0.818 − 0.219i)5-s + (2.40 − 0.922i)6-s + (2.69 − 0.867i)8-s + (−0.284 − 0.164i)9-s + (−0.930 − 0.755i)10-s + (−0.732 + 2.73i)11-s + (−3.05 − 1.99i)12-s + (1.00 − 1.00i)13-s + 1.54i·15-s + (−3.22 − 2.36i)16-s + (5.61 − 3.24i)17-s + (0.0481 + 0.461i)18-s + (5.37 − 1.43i)19-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.272 + 1.01i)3-s + (−0.310 + 0.950i)4-s + (0.366 − 0.0981i)5-s + (0.983 − 0.376i)6-s + (0.951 − 0.306i)8-s + (−0.0947 − 0.0546i)9-s + (−0.294 − 0.238i)10-s + (−0.220 + 0.823i)11-s + (−0.882 − 0.575i)12-s + (0.277 − 0.277i)13-s + 0.399i·15-s + (−0.807 − 0.590i)16-s + (1.36 − 0.785i)17-s + (0.0113 + 0.108i)18-s + (1.23 − 0.330i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.643 - 0.765i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.643 - 0.765i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.984445 + 0.458552i\)
\(L(\frac12)\) \(\approx\) \(0.984445 + 0.458552i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.830 + 1.14i)T \)
7 \( 1 \)
good3 \( 1 + (0.472 - 1.76i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.818 + 0.219i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.732 - 2.73i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.00 + 1.00i)T - 13iT^{2} \)
17 \( 1 + (-5.61 + 3.24i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.37 + 1.43i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.20 - 3.81i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.241 - 0.241i)T + 29iT^{2} \)
31 \( 1 + (1.69 + 2.93i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.00 - 3.73i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.88T + 41T^{2} \)
43 \( 1 + (-4.40 - 4.40i)T + 43iT^{2} \)
47 \( 1 + (4.72 - 8.18i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.39 + 1.17i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.84 + 0.494i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.60 - 13.4i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-11.9 - 3.18i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 + (-0.402 - 0.697i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.32 - 5.38i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.76 - 5.76i)T + 83iT^{2} \)
89 \( 1 + (-6.66 + 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.8iT - 97T^{2} \)
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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

−10.13378671392023960511582655205, −9.686167958462549716809111097666, −9.319200677974927074558903941725, −7.83671606218761258535366528954, −7.38445631471099042923388137784, −5.66695285750571464418109391784, −4.88207205869630728361386730292, −3.87138689244018041547722177043, −2.87339582426651789507565313158, −1.36261307199734342921346849342, 0.797178528225336569653521606386, 1.92577145424723194247982450072, 3.72841304922425852732107669643, 5.34520199129579953840612043257, 6.01163005497188969028499057488, 6.64879971364070749708224031638, 7.73272228881806331743945341514, 8.088069465781488994940289465773, 9.250447451791649341293620049549, 10.05037354324860669226617891963

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