↑

Properties

Label 2-28e2-16.13-c1-0-14
Degree $2$
Conductor $784$
Sign $-0.577 - 0.816i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.33 − 0.467i)2-s + (2.13 + 2.13i)3-s + (1.56 + 1.24i)4-s + (−0.545 + 0.545i)5-s + (−1.85 − 3.84i)6-s + (−1.50 − 2.39i)8-s + 6.10i·9-s + (0.983 − 0.473i)10-s + (−0.910 + 0.910i)11-s + (0.670 + 5.99i)12-s + (0.919 + 0.919i)13-s − 2.32·15-s + (0.883 + 3.90i)16-s − 7.91·17-s + (2.85 − 8.15i)18-s + (−1.30 − 1.30i)19-s + ⋯
L(s)  = 1  + (−0.943 − 0.330i)2-s + (1.23 + 1.23i)3-s + (0.781 + 0.624i)4-s + (−0.244 + 0.244i)5-s + (−0.755 − 1.57i)6-s + (−0.530 − 0.847i)8-s + 2.03i·9-s + (0.311 − 0.149i)10-s + (−0.274 + 0.274i)11-s + (0.193 + 1.73i)12-s + (0.254 + 0.254i)13-s − 0.601·15-s + (0.220 + 0.975i)16-s − 1.91·17-s + (0.673 − 1.92i)18-s + (−0.299 − 0.299i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.550700 + 1.06370i\)
\(L(\frac12)\) \(\approx\) \(0.550700 + 1.06370i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 + 0.467i)T \)
7 \( 1 \)
good3 \( 1 + (-2.13 - 2.13i)T + 3iT^{2} \)
5 \( 1 + (0.545 - 0.545i)T - 5iT^{2} \)
11 \( 1 + (0.910 - 0.910i)T - 11iT^{2} \)
13 \( 1 + (-0.919 - 0.919i)T + 13iT^{2} \)
17 \( 1 + 7.91T + 17T^{2} \)
19 \( 1 + (1.30 + 1.30i)T + 19iT^{2} \)
23 \( 1 - 3.84iT - 23T^{2} \)
29 \( 1 + (-5.25 - 5.25i)T + 29iT^{2} \)
31 \( 1 - 4.89T + 31T^{2} \)
37 \( 1 + (-0.938 + 0.938i)T - 37iT^{2} \)
41 \( 1 + 2.84iT - 41T^{2} \)
43 \( 1 + (-0.585 + 0.585i)T - 43iT^{2} \)
47 \( 1 + 5.72T + 47T^{2} \)
53 \( 1 + (-6.96 + 6.96i)T - 53iT^{2} \)
59 \( 1 + (6.54 - 6.54i)T - 59iT^{2} \)
61 \( 1 + (-4.67 - 4.67i)T + 61iT^{2} \)
67 \( 1 + (4.35 + 4.35i)T + 67iT^{2} \)
71 \( 1 + 1.99iT - 71T^{2} \)
73 \( 1 + 7.72iT - 73T^{2} \)
79 \( 1 - 9.26T + 79T^{2} \)
83 \( 1 + (-4.78 - 4.78i)T + 83iT^{2} \)
89 \( 1 - 2.12iT - 89T^{2} \)
97 \( 1 - 9.01T + 97T^{2} \)
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

−10.50775713261980698334753984917, −9.579067401116819627816904221423, −8.952083629133234518320718231133, −8.445031605354869598420004466894, −7.48293026265162347004076660278, −6.58199212977413954051131288259, −4.84056940552821318722982230212, −3.90265564132377356151532708550, −2.99379502581139051210507512252, −2.05605993736963304022880806319, 0.67258259938576826624874843171, 2.09802203029153356298591793645, 2.83200063952163528762097457172, 4.46789171899606956580474149554, 6.26389508735393068404164603804, 6.63538840990902271206206686424, 7.70861766300730112049528980492, 8.448746249280383512217780492058, 8.613279289711821579454008689942, 9.702286283817213184353900262205

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