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Properties

Label 2-28e2-28.27-c3-0-17
Degree $2$
Conductor $784$
Sign $-0.101 - 0.994i$
Analytic cond. $46.2574$
Root an. cond. $6.80128$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.34·3-s − 1.20i·5-s + 13.2·9-s + 32.0i·11-s + 59.4i·13-s − 7.63i·15-s − 0.0501i·17-s − 11.3·19-s + 75.2i·23-s + 123.·25-s − 87.3·27-s − 263.·29-s − 207.·31-s + 203. i·33-s + 199.·37-s + ⋯
L(s)  = 1  + 1.22·3-s − 0.107i·5-s + 0.489·9-s + 0.878i·11-s + 1.26i·13-s − 0.131i·15-s − 0.000715i·17-s − 0.137·19-s + 0.682i·23-s + 0.988·25-s − 0.622·27-s − 1.68·29-s − 1.20·31-s + 1.07i·33-s + 0.885·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.101 - 0.994i$
Analytic conductor: \(46.2574\)
Root analytic conductor: \(6.80128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :3/2),\ -0.101 - 0.994i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.377883302\)
\(L(\frac12)\) \(\approx\) \(2.377883302\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 - 6.34T + 27T^{2} \)
5 \( 1 + 1.20iT - 125T^{2} \)
11 \( 1 - 32.0iT - 1.33e3T^{2} \)
13 \( 1 - 59.4iT - 2.19e3T^{2} \)
17 \( 1 + 0.0501iT - 4.91e3T^{2} \)
19 \( 1 + 11.3T + 6.85e3T^{2} \)
23 \( 1 - 75.2iT - 1.21e4T^{2} \)
29 \( 1 + 263.T + 2.43e4T^{2} \)
31 \( 1 + 207.T + 2.97e4T^{2} \)
37 \( 1 - 199.T + 5.06e4T^{2} \)
41 \( 1 - 437. iT - 6.89e4T^{2} \)
43 \( 1 + 174. iT - 7.95e4T^{2} \)
47 \( 1 - 283.T + 1.03e5T^{2} \)
53 \( 1 - 7.41T + 1.48e5T^{2} \)
59 \( 1 + 38.6T + 2.05e5T^{2} \)
61 \( 1 - 276. iT - 2.26e5T^{2} \)
67 \( 1 - 965. iT - 3.00e5T^{2} \)
71 \( 1 - 580. iT - 3.57e5T^{2} \)
73 \( 1 + 607. iT - 3.89e5T^{2} \)
79 \( 1 - 827. iT - 4.93e5T^{2} \)
83 \( 1 - 1.20e3T + 5.71e5T^{2} \)
89 \( 1 - 312. iT - 7.04e5T^{2} \)
97 \( 1 + 1.24e3iT - 9.12e5T^{2} \)
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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

−9.711325646253627567588573425202, −9.286993581914086146711448313909, −8.567789792356457128745034520485, −7.54185022975016206090906505573, −6.98675150014580774061246623814, −5.68298384320296473397994705476, −4.46179137899539399831853341418, −3.65303965576777702046396681067, −2.45928265391868573680964359612, −1.59287518545746940304720434887, 0.50661464319940431458204452360, 2.12954087097636932985633866392, 3.11766823770402799508168696295, 3.78296242588791032911762271630, 5.23902538293704988031983061972, 6.12252233369566393752717279741, 7.41725162769738025307620829185, 8.001696034867946624330344439806, 8.862145421997063964001049191634, 9.369857995558402099624407362660

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