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Properties

Label 2-28e2-112.53-c1-0-52
Degree $2$
Conductor $784$
Sign $0.568 + 0.822i$
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  + (1.32 − 0.482i)2-s + (−0.331 − 1.23i)3-s + (1.53 − 1.28i)4-s + (−0.850 + 3.17i)5-s + (−1.03 − 1.48i)6-s + (1.41 − 2.44i)8-s + (1.17 − 0.679i)9-s + (0.402 + 4.62i)10-s + (−0.410 + 0.110i)11-s + (−2.09 − 1.47i)12-s + (3.70 − 3.70i)13-s + 4.21·15-s + (0.704 − 3.93i)16-s + (−1.35 + 2.34i)17-s + (1.23 − 1.47i)18-s + (3.41 + 0.914i)19-s + ⋯
L(s)  = 1  + (0.939 − 0.341i)2-s + (−0.191 − 0.714i)3-s + (0.766 − 0.641i)4-s + (−0.380 + 1.41i)5-s + (−0.423 − 0.606i)6-s + (0.501 − 0.865i)8-s + (0.392 − 0.226i)9-s + (0.127 + 1.46i)10-s + (−0.123 + 0.0331i)11-s + (−0.605 − 0.425i)12-s + (1.02 − 1.02i)13-s + 1.08·15-s + (0.176 − 0.984i)16-s + (−0.328 + 0.568i)17-s + (0.291 − 0.346i)18-s + (0.782 + 0.209i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.568 + 0.822i$
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: \((2,\ 784,\ (\ :1/2),\ 0.568 + 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.34265 - 1.22939i\)
\(L(\frac12)\) \(\approx\) \(2.34265 - 1.22939i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 + 0.482i)T \)
7 \( 1 \)
good3 \( 1 + (0.331 + 1.23i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.850 - 3.17i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.410 - 0.110i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.70 + 3.70i)T - 13iT^{2} \)
17 \( 1 + (1.35 - 2.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.41 - 0.914i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.36 + 2.51i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.464 + 0.464i)T - 29iT^{2} \)
31 \( 1 + (-3.87 + 6.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.40 - 5.24i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 10.5iT - 41T^{2} \)
43 \( 1 + (-8.35 - 8.35i)T + 43iT^{2} \)
47 \( 1 + (5.11 + 8.86i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (12.3 - 3.30i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (9.82 - 2.63i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.86 + 1.84i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.0652 - 0.243i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 6.38iT - 71T^{2} \)
73 \( 1 + (-1.22 - 0.706i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.14 + 7.17i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.84 - 2.84i)T - 83iT^{2} \)
89 \( 1 + (-0.471 + 0.272i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.67T + 97T^{2} \)
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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

−10.49299609966949451707518857296, −9.709966643490603673072043974309, −8.014108968488176728451567664870, −7.38479188144217982325902844908, −6.32573097188746448849106700773, −6.15919577938386305360818515878, −4.60641370732434095016046047911, −3.44967349469974527593103200266, −2.77069173401464692974682501568, −1.24437372956462584247881890145, 1.53615336832808189676057371124, 3.37406393310641600865077034454, 4.35461916712525515178170783612, 4.86202022674944110357657036452, 5.63725215491257089879789268228, 6.89257170885393064269495382571, 7.73474355044448651806450216492, 8.858460720253205959276630847659, 9.291147541562356452562509394820, 10.74382837316236649046682239132

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