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Properties

Label 2-28e2-112.53-c1-0-58
Degree $2$
Conductor $784$
Sign $0.998 - 0.0626i$
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 − 0.116i)2-s + (0.312 + 1.16i)3-s + (1.97 − 0.327i)4-s + (0.262 − 0.980i)5-s + (0.575 + 1.60i)6-s + (2.74 − 0.690i)8-s + (1.33 − 0.772i)9-s + (0.256 − 1.41i)10-s + (−2.36 + 0.635i)11-s + (0.997 + 2.19i)12-s + (2.65 − 2.65i)13-s + 1.22·15-s + (3.78 − 1.29i)16-s + (0.509 − 0.881i)17-s + (1.79 − 1.24i)18-s + (−0.0936 − 0.0250i)19-s + ⋯
L(s)  = 1  + (0.996 − 0.0820i)2-s + (0.180 + 0.672i)3-s + (0.986 − 0.163i)4-s + (0.117 − 0.438i)5-s + (0.234 + 0.655i)6-s + (0.969 − 0.243i)8-s + (0.445 − 0.257i)9-s + (0.0811 − 0.446i)10-s + (−0.714 + 0.191i)11-s + (0.287 + 0.634i)12-s + (0.737 − 0.737i)13-s + 0.316·15-s + (0.946 − 0.322i)16-s + (0.123 − 0.213i)17-s + (0.423 − 0.293i)18-s + (−0.0214 − 0.00575i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.998 - 0.0626i$
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: \((2,\ 784,\ (\ :1/2),\ 0.998 - 0.0626i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.27190 + 0.102553i\)
\(L(\frac12)\) \(\approx\) \(3.27190 + 0.102553i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.116i)T \)
7 \( 1 \)
good3 \( 1 + (-0.312 - 1.16i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.262 + 0.980i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.36 - 0.635i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-2.65 + 2.65i)T - 13iT^{2} \)
17 \( 1 + (-0.509 + 0.881i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0936 + 0.0250i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.67 - 0.965i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.05 - 5.05i)T - 29iT^{2} \)
31 \( 1 + (4.28 - 7.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.06 + 7.71i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 8.51iT - 41T^{2} \)
43 \( 1 + (4.47 + 4.47i)T + 43iT^{2} \)
47 \( 1 + (6.02 + 10.4i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.42 - 0.381i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (6.96 - 1.86i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.42 + 1.18i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.907 - 3.38i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 5.43iT - 71T^{2} \)
73 \( 1 + (-7.34 - 4.23i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.433 + 0.751i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.44 + 5.44i)T - 83iT^{2} \)
89 \( 1 + (-3.93 + 2.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 16.4T + 97T^{2} \)
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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

−10.51748010851719834336726701919, −9.621111482334147332873115312214, −8.660912380437285242423857200410, −7.57553892870616608077747711759, −6.70125138160333286544778900634, −5.45305914644131868250873713726, −4.99353309687155995309557245783, −3.82741163945073909784510100178, −3.10941964066064380804826334207, −1.52501506943628373253936706502, 1.69126367009939234938540605752, 2.65603157807735266491138065977, 3.86064863665308237556786035273, 4.84103665079545321833850202881, 6.07444464292736855794709912846, 6.58574069203249906011518959998, 7.63613736830069312128542268759, 8.113375511236834123374377468817, 9.551206655716848005183499208910, 10.61171754981044908946722477686

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