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Properties

Label 2-28e2-112.53-c1-0-32
Degree $2$
Conductor $784$
Sign $0.999 + 0.0147i$
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  + (−1.14 − 0.830i)2-s + (0.261 + 0.977i)3-s + (0.621 + 1.90i)4-s + (−0.317 + 1.18i)5-s + (0.511 − 1.33i)6-s + (0.867 − 2.69i)8-s + (1.71 − 0.988i)9-s + (1.34 − 1.09i)10-s + (4.06 − 1.08i)11-s + (−1.69 + 1.10i)12-s + (2.02 − 2.02i)13-s − 1.24·15-s + (−3.22 + 2.36i)16-s + (0.132 − 0.229i)17-s + (−2.78 − 0.290i)18-s + (−6.20 − 1.66i)19-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.151 + 0.564i)3-s + (0.310 + 0.950i)4-s + (−0.142 + 0.530i)5-s + (0.208 − 0.545i)6-s + (0.306 − 0.951i)8-s + (0.570 − 0.329i)9-s + (0.426 − 0.345i)10-s + (1.22 − 0.328i)11-s + (−0.489 + 0.318i)12-s + (0.560 − 0.560i)13-s − 0.320·15-s + (−0.807 + 0.590i)16-s + (0.0320 − 0.0555i)17-s + (−0.655 − 0.0683i)18-s + (−1.42 − 0.381i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.999 + 0.0147i$
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: \((2,\ 784,\ (\ :1/2),\ 0.999 + 0.0147i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22641 - 0.00904751i\)
\(L(\frac12)\) \(\approx\) \(1.22641 - 0.00904751i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.14 + 0.830i)T \)
7 \( 1 \)
good3 \( 1 + (-0.261 - 0.977i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.317 - 1.18i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-4.06 + 1.08i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-2.02 + 2.02i)T - 13iT^{2} \)
17 \( 1 + (-0.132 + 0.229i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.20 + 1.66i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.33 + 0.773i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.328 - 0.328i)T - 29iT^{2} \)
31 \( 1 + (-3.02 + 5.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.43 + 9.08i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 11.0iT - 41T^{2} \)
43 \( 1 + (-3.38 - 3.38i)T + 43iT^{2} \)
47 \( 1 + (-1.56 - 2.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.588 - 0.157i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (6.31 - 1.69i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-6.64 - 1.78i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.22 - 4.56i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.03iT - 71T^{2} \)
73 \( 1 + (-12.8 - 7.40i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.29 + 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.715 + 0.715i)T - 83iT^{2} \)
89 \( 1 + (-9.51 + 5.49i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.2T + 97T^{2} \)
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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

−10.32538398032327604350786257377, −9.404520288949390117538815714885, −8.891687725906149510686672223050, −7.942453852940666058495756276276, −6.88396780264297455479490950096, −6.22698107082898074853862067739, −4.37902286097052347760726220481, −3.72874053189700524550784112092, −2.68165659304051860367199858160, −1.09080646393959547958695122767, 1.13421729419213883257552096824, 2.05043778373961596631419483145, 4.06728317083493552422204171481, 4.99174734438966346758530813687, 6.47486801144870909135205397469, 6.69667226447445290761964923226, 7.82005844605523365002530912876, 8.610487660656934038948545021754, 9.131802440709617314803594648097, 10.18522101931428747508142763198

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