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Properties

Label 2-28e2-112.109-c1-0-22
Degree $2$
Conductor $784$
Sign $0.370 - 0.928i$
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  + (−1.18 + 0.768i)2-s + (−1.70 + 0.456i)3-s + (0.819 − 1.82i)4-s + (0.627 + 0.168i)5-s + (1.67 − 1.85i)6-s + (0.428 + 2.79i)8-s + (0.0942 − 0.0543i)9-s + (−0.874 + 0.282i)10-s + (−1.00 − 3.75i)11-s + (−0.563 + 3.48i)12-s + (1.94 + 1.94i)13-s − 1.14·15-s + (−2.65 − 2.99i)16-s + (−1.88 + 3.27i)17-s + (−0.0700 + 0.136i)18-s + (−0.190 + 0.710i)19-s + ⋯
L(s)  = 1  + (−0.839 + 0.543i)2-s + (−0.983 + 0.263i)3-s + (0.409 − 0.912i)4-s + (0.280 + 0.0752i)5-s + (0.682 − 0.755i)6-s + (0.151 + 0.988i)8-s + (0.0314 − 0.0181i)9-s + (−0.276 + 0.0893i)10-s + (−0.303 − 1.13i)11-s + (−0.162 + 1.00i)12-s + (0.538 + 0.538i)13-s − 0.295·15-s + (−0.664 − 0.747i)16-s + (−0.457 + 0.793i)17-s + (−0.0165 + 0.0322i)18-s + (−0.0436 + 0.163i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.370 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.370 - 0.928i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.370 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.552091 + 0.374319i\)
\(L(\frac12)\) \(\approx\) \(0.552091 + 0.374319i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.18 - 0.768i)T \)
7 \( 1 \)
good3 \( 1 + (1.70 - 0.456i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.627 - 0.168i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.00 + 3.75i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.94 - 1.94i)T + 13iT^{2} \)
17 \( 1 + (1.88 - 3.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.190 - 0.710i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-7.62 + 4.40i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.41 - 1.41i)T + 29iT^{2} \)
31 \( 1 + (-0.927 + 1.60i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.20 - 0.592i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 11.2iT - 41T^{2} \)
43 \( 1 + (-2.22 + 2.22i)T - 43iT^{2} \)
47 \( 1 + (-4.01 - 6.95i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.36 - 5.09i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.61 - 9.74i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.03 + 11.3i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.71 + 0.726i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 11.0iT - 71T^{2} \)
73 \( 1 + (-10.3 - 5.94i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.64 - 8.04i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.19 + 4.19i)T + 83iT^{2} \)
89 \( 1 + (12.2 - 7.07i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.415T + 97T^{2} \)
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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

−10.61243378494311341514734091399, −9.616172691587349756353596536260, −8.647472669753931700159342433548, −8.109389530944991453761849490109, −6.72046069675116761401758373418, −6.15846962256826448704083414192, −5.48286342127512804358584378378, −4.40883840384518656242954331911, −2.63243905540533372073464161269, −0.943413823564972510279178156246, 0.68269995848066544840749508041, 2.08553273167663398243312441938, 3.37671803887268273582418658685, 4.82633500356805021737177652996, 5.75445674886361103045170335574, 6.92587446087037626739363102041, 7.39141068459722817795396648424, 8.640024983994896265102011687380, 9.392565966864137862123766727305, 10.20306873334862477514039966597

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