↑

Properties

Label 2-28e2-112.93-c1-0-56
Degree $2$
Conductor $784$
Sign $-0.948 + 0.316i$
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  + (0.114 − 1.40i)2-s + (−0.509 + 1.90i)3-s + (−1.97 − 0.323i)4-s + (−0.792 − 2.95i)5-s + (2.62 + 0.935i)6-s + (−0.681 + 2.74i)8-s + (−0.754 − 0.435i)9-s + (−4.25 + 0.778i)10-s + (4.22 + 1.13i)11-s + (1.61 − 3.58i)12-s + (−1.75 − 1.75i)13-s + 6.02·15-s + (3.79 + 1.27i)16-s + (−2.60 − 4.50i)17-s + (−0.700 + 1.01i)18-s + (−1.16 + 0.311i)19-s + ⋯
L(s)  = 1  + (0.0810 − 0.996i)2-s + (−0.294 + 1.09i)3-s + (−0.986 − 0.161i)4-s + (−0.354 − 1.32i)5-s + (1.06 + 0.381i)6-s + (−0.240 + 0.970i)8-s + (−0.251 − 0.145i)9-s + (−1.34 + 0.246i)10-s + (1.27 + 0.341i)11-s + (0.467 − 1.03i)12-s + (−0.486 − 0.486i)13-s + 1.55·15-s + (0.947 + 0.318i)16-s + (−0.631 − 1.09i)17-s + (−0.165 + 0.238i)18-s + (−0.266 + 0.0715i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.948 + 0.316i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (765, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.948 + 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.116669 - 0.718884i\)
\(L(\frac12)\) \(\approx\) \(0.116669 - 0.718884i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.114 + 1.40i)T \)
7 \( 1 \)
good3 \( 1 + (0.509 - 1.90i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (0.792 + 2.95i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-4.22 - 1.13i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.75 + 1.75i)T + 13iT^{2} \)
17 \( 1 + (2.60 + 4.50i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.16 - 0.311i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (5.33 + 3.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.24 + 6.24i)T + 29iT^{2} \)
31 \( 1 + (1.39 + 2.40i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.50 - 5.61i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 6.32iT - 41T^{2} \)
43 \( 1 + (-3.05 + 3.05i)T - 43iT^{2} \)
47 \( 1 + (-1.80 + 3.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.21 + 1.93i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (9.74 + 2.61i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.42 + 0.380i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (0.353 - 1.32i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 10.0iT - 71T^{2} \)
73 \( 1 + (13.1 - 7.58i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.30 + 5.72i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.41 - 7.41i)T + 83iT^{2} \)
89 \( 1 + (-2.82 - 1.63i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.66T + 97T^{2} \)
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

−9.791295907499733513767512622513, −9.370139279882484353450858293392, −8.680788535317317452323614073327, −7.60967663394576078812608406077, −5.94565684618941351365022906308, −4.87534077594004011232521619346, −4.42574526547702189602903201103, −3.70348327487795616291169381151, −2.00114853434973273109651908606, −0.37713840906264042465861182419, 1.69400098507349242696558203144, 3.45242221932830674643155791356, 4.30969492049564013547501830274, 6.06612704692712676175766650955, 6.31167614648058881070618286748, 7.22401570845639116825690680546, 7.59424321584103746274102066243, 8.768764847002088128112840430531, 9.622370854742437927179867503437, 10.79158839030708511232300478793

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