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Properties

Label 2-28e2-28.27-c3-0-19
Degree $2$
Conductor $784$
Sign $-0.810 - 0.585i$
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  + 5.68·3-s + 20.4i·5-s + 5.37·9-s + 27.4i·11-s + 33.2i·13-s + 116. i·15-s + 51.6i·17-s + 123.·19-s − 203. i·23-s − 291.·25-s − 123.·27-s − 27.1·29-s + 132.·31-s + 155. i·33-s − 98.1·37-s + ⋯
L(s)  = 1  + 1.09·3-s + 1.82i·5-s + 0.198·9-s + 0.751i·11-s + 0.709i·13-s + 1.99i·15-s + 0.736i·17-s + 1.48·19-s − 1.84i·23-s − 2.33·25-s − 0.877·27-s − 0.173·29-s + 0.768·31-s + 0.822i·33-s − 0.435·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.810 - 0.585i$
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: \((2,\ 784,\ (\ :3/2),\ -0.810 - 0.585i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.398388997\)
\(L(\frac12)\) \(\approx\) \(2.398388997\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.68T + 27T^{2} \)
5 \( 1 - 20.4iT - 125T^{2} \)
11 \( 1 - 27.4iT - 1.33e3T^{2} \)
13 \( 1 - 33.2iT - 2.19e3T^{2} \)
17 \( 1 - 51.6iT - 4.91e3T^{2} \)
19 \( 1 - 123.T + 6.85e3T^{2} \)
23 \( 1 + 203. iT - 1.21e4T^{2} \)
29 \( 1 + 27.1T + 2.43e4T^{2} \)
31 \( 1 - 132.T + 2.97e4T^{2} \)
37 \( 1 + 98.1T + 5.06e4T^{2} \)
41 \( 1 - 475. iT - 6.89e4T^{2} \)
43 \( 1 - 288. iT - 7.95e4T^{2} \)
47 \( 1 + 568.T + 1.03e5T^{2} \)
53 \( 1 + 133.T + 1.48e5T^{2} \)
59 \( 1 - 307.T + 2.05e5T^{2} \)
61 \( 1 + 735. iT - 2.26e5T^{2} \)
67 \( 1 + 130. iT - 3.00e5T^{2} \)
71 \( 1 + 708. iT - 3.57e5T^{2} \)
73 \( 1 - 562. iT - 3.89e5T^{2} \)
79 \( 1 - 787. iT - 4.93e5T^{2} \)
83 \( 1 + 300.T + 5.71e5T^{2} \)
89 \( 1 + 1.13e3iT - 7.04e5T^{2} \)
97 \( 1 - 234. iT - 9.12e5T^{2} \)
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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

−9.967052855777111985701868457916, −9.657120603591505308585180860370, −8.375531379377145953319173145615, −7.72064592458441033405616701296, −6.80871430859305924876251878159, −6.23350110338908790396455033354, −4.61080554349916336036136009518, −3.43238582606722185264022450706, −2.80889809195576425058273208908, −1.88409440612352807958282211766, 0.53194558327664332965958794828, 1.58452525257216373201066525346, 3.04358362815084381765324549980, 3.86263189320172547801612041465, 5.27545722173995647262553175080, 5.54684503159688053851022727252, 7.38985912197011915786578869106, 8.040050675479643875267182537031, 8.772809362426454368261687808192, 9.290723382544155404152369320025

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