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Properties

Label 2-28e2-112.53-c1-0-74
Degree $2$
Conductor $784$
Sign $-0.952 - 0.305i$
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  + (1.01 − 0.988i)2-s + (−0.430 − 1.60i)3-s + (0.0448 − 1.99i)4-s + (0.522 − 1.94i)5-s + (−2.02 − 1.20i)6-s + (−1.93 − 2.06i)8-s + (0.196 − 0.113i)9-s + (−1.39 − 2.48i)10-s + (−6.09 + 1.63i)11-s + (−3.23 + 0.789i)12-s + (−1.13 + 1.13i)13-s − 3.35·15-s + (−3.99 − 0.179i)16-s + (0.960 − 1.66i)17-s + (0.0866 − 0.309i)18-s + (6.09 + 1.63i)19-s + ⋯
L(s)  = 1  + (0.714 − 0.699i)2-s + (−0.248 − 0.928i)3-s + (0.0224 − 0.999i)4-s + (0.233 − 0.871i)5-s + (−0.827 − 0.490i)6-s + (−0.682 − 0.730i)8-s + (0.0655 − 0.0378i)9-s + (−0.442 − 0.786i)10-s + (−1.83 + 0.492i)11-s + (−0.933 + 0.227i)12-s + (−0.314 + 0.314i)13-s − 0.867·15-s + (−0.998 − 0.0448i)16-s + (0.233 − 0.403i)17-s + (0.0204 − 0.0729i)18-s + (1.39 + 0.374i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.952 - 0.305i$
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: \((2,\ 784,\ (\ :1/2),\ -0.952 - 0.305i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.270487 + 1.72669i\)
\(L(\frac12)\) \(\approx\) \(0.270487 + 1.72669i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.01 + 0.988i)T \)
7 \( 1 \)
good3 \( 1 + (0.430 + 1.60i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.522 + 1.94i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (6.09 - 1.63i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.13 - 1.13i)T - 13iT^{2} \)
17 \( 1 + (-0.960 + 1.66i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.09 - 1.63i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.924 + 0.533i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.08 + 5.08i)T - 29iT^{2} \)
31 \( 1 + (-0.198 + 0.343i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0877 - 0.327i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 7.26iT - 41T^{2} \)
43 \( 1 + (-1.75 - 1.75i)T + 43iT^{2} \)
47 \( 1 + (-1.08 - 1.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.415 - 0.111i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (8.67 - 2.32i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (10.1 + 2.72i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.10 + 7.87i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 2.78iT - 71T^{2} \)
73 \( 1 + (-2.05 - 1.18i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.10 - 8.84i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.17 - 1.17i)T - 83iT^{2} \)
89 \( 1 + (-11.0 + 6.37i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.21T + 97T^{2} \)
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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

−9.930469800406886173722987290004, −9.258263808226518482344446261094, −7.895774561509381352437340892723, −7.22538665573485160009965672432, −6.04388331579125610300316530511, −5.21372157063409179591101951493, −4.56811626640350446623485668955, −2.98585839414557489655878954368, −1.91036224763615977391417985412, −0.70869910119194367421605615802, 2.76251964482643351508118758421, 3.34182172608848012744836075053, 4.80066167323684451473483930296, 5.25112536091729357693522883499, 6.19693338587747512655156440351, 7.33643373682798690267787831780, 7.87174892301084545201863801691, 9.046981291921608332068342106857, 10.20528209011568304918532418793, 10.60470428374859049431347209825

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