↑

Properties

Label 2-760-8.5-c1-0-68
Degree $2$
Conductor $760$
Sign $-0.878 - 0.477i$
Analytic cond. $6.06863$
Root an. cond. $2.46345$
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  + (0.899 − 1.09i)2-s − 1.00i·3-s + (−0.381 − 1.96i)4-s − i·5-s + (−1.09 − 0.904i)6-s − 4.51·7-s + (−2.48 − 1.35i)8-s + 1.98·9-s + (−1.09 − 0.899i)10-s − 2.44i·11-s + (−1.97 + 0.383i)12-s + 1.51i·13-s + (−4.06 + 4.92i)14-s − 1.00·15-s + (−3.70 + 1.49i)16-s + 1.39·17-s + ⋯
L(s)  = 1  + (0.636 − 0.771i)2-s − 0.580i·3-s + (−0.190 − 0.981i)4-s − 0.447i·5-s + (−0.448 − 0.369i)6-s − 1.70·7-s + (−0.878 − 0.477i)8-s + 0.662·9-s + (−0.345 − 0.284i)10-s − 0.738i·11-s + (−0.570 + 0.110i)12-s + 0.419i·13-s + (−1.08 + 1.31i)14-s − 0.259·15-s + (−0.927 + 0.374i)16-s + 0.338·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 - 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.878 - 0.477i$
Analytic conductor: \(6.06863\)
Root analytic conductor: \(2.46345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 760,\ (\ :1/2),\ -0.878 - 0.477i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.305988 + 1.20418i\)
\(L(\frac12)\) \(\approx\) \(0.305988 + 1.20418i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.899 + 1.09i)T \)
5 \( 1 + iT \)
19 \( 1 + iT \)
good3 \( 1 + 1.00iT - 3T^{2} \)
7 \( 1 + 4.51T + 7T^{2} \)
11 \( 1 + 2.44iT - 11T^{2} \)
13 \( 1 - 1.51iT - 13T^{2} \)
17 \( 1 - 1.39T + 17T^{2} \)
23 \( 1 + 7.19T + 23T^{2} \)
29 \( 1 - 3.84iT - 29T^{2} \)
31 \( 1 - 3.46T + 31T^{2} \)
37 \( 1 - 0.407iT - 37T^{2} \)
41 \( 1 + 7.23T + 41T^{2} \)
43 \( 1 + 5.25iT - 43T^{2} \)
47 \( 1 - 5.36T + 47T^{2} \)
53 \( 1 + 12.8iT - 53T^{2} \)
59 \( 1 + 4.37iT - 59T^{2} \)
61 \( 1 + 9.25iT - 61T^{2} \)
67 \( 1 + 0.564iT - 67T^{2} \)
71 \( 1 + 3.85T + 71T^{2} \)
73 \( 1 - 7.95T + 73T^{2} \)
79 \( 1 - 12.8T + 79T^{2} \)
83 \( 1 + 7.43iT - 83T^{2} \)
89 \( 1 - 6.13T + 89T^{2} \)
97 \( 1 + 9.57T + 97T^{2} \)
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

−9.894802547652530555009725535898, −9.325569865927854538086453405285, −8.232684320011402807267187954351, −6.81580669126157266135646988211, −6.34457300018715770266855132548, −5.33762611852788128936228869427, −4.03537366751144013605067914473, −3.27473446568753737064037481817, −1.96400685183300680313091319673, −0.49369659697528285947344070171, 2.66975661764247067249927993284, 3.68025919362280851541715527605, 4.34859103592156185923712370316, 5.66176508429257023654306289370, 6.41182447359952863732606454583, 7.16416658319155268101517519844, 8.007366031576683467505772334569, 9.326744671882292578650336771665, 9.883182886347134907075961273927, 10.52958627394641774445073089470

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