↑

Properties

Label 2-28e2-28.27-c3-0-32
Degree $2$
Conductor $784$
Sign $0.944 + 0.327i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5.19i·5-s − 26·9-s + 25.9i·11-s − 69.2i·13-s + 5.19i·15-s + 36.3i·17-s − 17·19-s − 140. i·23-s + 98·25-s − 53·27-s + 90·29-s + 17·31-s + 25.9i·33-s + 199·37-s + ⋯
L(s)  = 1  + 0.192·3-s + 0.464i·5-s − 0.962·9-s + 0.712i·11-s − 1.47i·13-s + 0.0894i·15-s + 0.518i·17-s − 0.205·19-s − 1.27i·23-s + 0.784·25-s − 0.377·27-s + 0.576·29-s + 0.0984·31-s + 0.137i·33-s + 0.884·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.944 + 0.327i$
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: \((2,\ 784,\ (\ :3/2),\ 0.944 + 0.327i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.822173291\)
\(L(\frac12)\) \(\approx\) \(1.822173291\)
\(L(\frac{5}{2})\) 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 \( 1 - T + 27T^{2} \)
5 \( 1 - 5.19iT - 125T^{2} \)
11 \( 1 - 25.9iT - 1.33e3T^{2} \)
13 \( 1 + 69.2iT - 2.19e3T^{2} \)
17 \( 1 - 36.3iT - 4.91e3T^{2} \)
19 \( 1 + 17T + 6.85e3T^{2} \)
23 \( 1 + 140. iT - 1.21e4T^{2} \)
29 \( 1 - 90T + 2.43e4T^{2} \)
31 \( 1 - 17T + 2.97e4T^{2} \)
37 \( 1 - 199T + 5.06e4T^{2} \)
41 \( 1 - 187. iT - 6.89e4T^{2} \)
43 \( 1 + 252. iT - 7.95e4T^{2} \)
47 \( 1 - 567T + 1.03e5T^{2} \)
53 \( 1 + 333T + 1.48e5T^{2} \)
59 \( 1 - 801T + 2.05e5T^{2} \)
61 \( 1 - 358. iT - 2.26e5T^{2} \)
67 \( 1 + 216. iT - 3.00e5T^{2} \)
71 \( 1 - 488. iT - 3.57e5T^{2} \)
73 \( 1 - 403. iT - 3.89e5T^{2} \)
79 \( 1 + 1.19e3iT - 4.93e5T^{2} \)
83 \( 1 + 468T + 5.71e5T^{2} \)
89 \( 1 - 192. iT - 7.04e5T^{2} \)
97 \( 1 + 1.39e3iT - 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.08625448033488253627204241876, −8.825363424977135245613830119902, −8.241639774249464599270659743391, −7.31760436815603365189497744577, −6.32192105446161126838520262031, −5.49225328845156933255708393884, −4.38147885632415014528225395214, −3.10844742437699930701417144332, −2.39558192399361083250023114192, −0.63427968983856953943614850514, 0.875414153812925214271839062980, 2.29366961584020614908430557906, 3.41662945852114830799340753860, 4.52519250660219128972701369103, 5.51381940506546798849812377841, 6.39553546636280689912845951909, 7.41101050693824340400889151760, 8.457543804569861884558321297974, 9.008547514151542791740707863192, 9.709528938659603298324206076174

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