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Properties

Label 2-28e2-112.53-c1-0-71
Degree $2$
Conductor $784$
Sign $-0.745 - 0.666i$
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  + (−1.29 + 0.560i)2-s + (−0.752 − 2.80i)3-s + (1.37 − 1.45i)4-s + (0.998 − 3.72i)5-s + (2.55 + 3.22i)6-s + (−0.964 + 2.65i)8-s + (−4.71 + 2.72i)9-s + (0.792 + 5.39i)10-s + (−1.25 + 0.336i)11-s + (−5.11 − 2.75i)12-s + (−1.12 + 1.12i)13-s − 11.2·15-s + (−0.238 − 3.99i)16-s + (0.754 − 1.30i)17-s + (4.59 − 6.17i)18-s + (−1.99 − 0.535i)19-s + ⋯
L(s)  = 1  + (−0.918 + 0.396i)2-s + (−0.434 − 1.62i)3-s + (0.685 − 0.727i)4-s + (0.446 − 1.66i)5-s + (1.04 + 1.31i)6-s + (−0.340 + 0.940i)8-s + (−1.57 + 0.907i)9-s + (0.250 + 1.70i)10-s + (−0.378 + 0.101i)11-s + (−1.47 − 0.795i)12-s + (−0.312 + 0.312i)13-s − 2.89·15-s + (−0.0596 − 0.998i)16-s + (0.182 − 0.316i)17-s + (1.08 − 1.45i)18-s + (−0.458 − 0.122i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.745 - 0.666i$
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: \((2,\ 784,\ (\ :1/2),\ -0.745 - 0.666i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.174716 + 0.457290i\)
\(L(\frac12)\) \(\approx\) \(0.174716 + 0.457290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 - 0.560i)T \)
7 \( 1 \)
good3 \( 1 + (0.752 + 2.80i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.998 + 3.72i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.25 - 0.336i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.12 - 1.12i)T - 13iT^{2} \)
17 \( 1 + (-0.754 + 1.30i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.99 + 0.535i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.13 - 2.38i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.10 + 4.10i)T - 29iT^{2} \)
31 \( 1 + (2.05 - 3.55i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.605 - 2.26i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 7.45iT - 41T^{2} \)
43 \( 1 + (-5.68 - 5.68i)T + 43iT^{2} \)
47 \( 1 + (1.79 + 3.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.923 + 0.247i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.55 - 0.416i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.39 - 1.17i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.558 + 2.08i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 + (12.5 + 7.23i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.889 - 1.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.16 - 7.16i)T - 83iT^{2} \)
89 \( 1 + (-7.32 + 4.22i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−9.504865976670978383209018768433, −8.717924842176001312992403663913, −8.018228491752184648664075274781, −7.35696743534597802209639863197, −6.32559032355775570313519905717, −5.65753405110308383027815797447, −4.79506067321049797648134362159, −2.26827290948581196532862271345, −1.44117574336422306451240300473, −0.34535260417608358046669672292, 2.41946216314920893968255665493, 3.26047229990003837047057156939, 4.18965648416962278094436696326, 5.67519677645968085951525414115, 6.42825193023361083776095463609, 7.46524807616392467555236026193, 8.527568381369232225199135392935, 9.550507707335377552388365021574, 10.22166963691382532244753739960, 10.49586252993472636597496245974

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