↑

Properties

Label 2-760-8.5-c1-0-36
Degree $2$
Conductor $760$
Sign $0.994 + 0.106i$
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  + (−0.750 + 1.19i)2-s − 1.80i·3-s + (−0.874 − 1.79i)4-s + i·5-s + (2.15 + 1.35i)6-s + 4.97·7-s + (2.81 + 0.301i)8-s − 0.244·9-s + (−1.19 − 0.750i)10-s − 5.55i·11-s + (−3.23 + 1.57i)12-s + 5.33i·13-s + (−3.73 + 5.96i)14-s + 1.80·15-s + (−2.47 + 3.14i)16-s − 0.586·17-s + ⋯
L(s)  = 1  + (−0.530 + 0.847i)2-s − 1.03i·3-s + (−0.437 − 0.899i)4-s + 0.447i·5-s + (0.881 + 0.551i)6-s + 1.88·7-s + (0.994 + 0.106i)8-s − 0.0814·9-s + (−0.379 − 0.237i)10-s − 1.67i·11-s + (−0.935 + 0.454i)12-s + 1.47i·13-s + (−0.997 + 1.59i)14-s + 0.465·15-s + (−0.617 + 0.786i)16-s − 0.142·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.994 + 0.106i$
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: \((2,\ 760,\ (\ :1/2),\ 0.994 + 0.106i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43442 - 0.0766032i\)
\(L(\frac12)\) \(\approx\) \(1.43442 - 0.0766032i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.750 - 1.19i)T \)
5 \( 1 - iT \)
19 \( 1 - iT \)
good3 \( 1 + 1.80iT - 3T^{2} \)
7 \( 1 - 4.97T + 7T^{2} \)
11 \( 1 + 5.55iT - 11T^{2} \)
13 \( 1 - 5.33iT - 13T^{2} \)
17 \( 1 + 0.586T + 17T^{2} \)
23 \( 1 - 1.77T + 23T^{2} \)
29 \( 1 - 7.19iT - 29T^{2} \)
31 \( 1 + 2.10T + 31T^{2} \)
37 \( 1 + 3.39iT - 37T^{2} \)
41 \( 1 - 5.88T + 41T^{2} \)
43 \( 1 + 2.50iT - 43T^{2} \)
47 \( 1 - 6.94T + 47T^{2} \)
53 \( 1 + 11.8iT - 53T^{2} \)
59 \( 1 + 5.63iT - 59T^{2} \)
61 \( 1 - 2.34iT - 61T^{2} \)
67 \( 1 - 12.1iT - 67T^{2} \)
71 \( 1 + 6.57T + 71T^{2} \)
73 \( 1 + 1.31T + 73T^{2} \)
79 \( 1 + 0.202T + 79T^{2} \)
83 \( 1 + 8.37iT - 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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

−10.42210643582348019827151206298, −8.927201883289107393866055535293, −8.543499538473168286650359389281, −7.59986816593401098628202757302, −7.06941796963800417671679599220, −6.13990360587783118986427147190, −5.23772070239845637352497317029, −4.10678810162614372790523667437, −2.04866558517961946796977439017, −1.13549560733304056046898369341, 1.32225839440192098620038276264, 2.51607449334890693920808477555, 4.14678860304199986320708806844, 4.61111360795213450378197554947, 5.33769384809842170507039004648, 7.48409625355587028866122468507, 7.87069400086105616449470635487, 8.895900328358064932199956416021, 9.621116886124639451051875697513, 10.44320827859013889639967549534

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