↑

Properties

Label 2-1300-65.2-c1-0-16
Degree $2$
Conductor $1300$
Sign $0.492 + 0.870i$
Analytic cond. $10.3805$
Root an. cond. $3.22188$
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  + (−0.243 − 0.0651i)3-s + (4.30 + 2.48i)7-s + (−2.54 − 1.46i)9-s + (1.07 − 4.01i)11-s + (−3.20 − 1.65i)13-s + (−1.65 − 6.16i)17-s + (1.01 − 0.270i)19-s + (−0.885 − 0.885i)21-s + (−0.195 + 0.730i)23-s + (1.05 + 1.05i)27-s + (6.59 − 3.80i)29-s + (2.17 − 2.17i)31-s + (−0.522 + 0.905i)33-s + (−5.36 + 3.09i)37-s + (0.670 + 0.610i)39-s + ⋯
L(s)  = 1  + (−0.140 − 0.0375i)3-s + (1.62 + 0.940i)7-s + (−0.847 − 0.489i)9-s + (0.324 − 1.21i)11-s + (−0.888 − 0.459i)13-s + (−0.400 − 1.49i)17-s + (0.231 − 0.0621i)19-s + (−0.193 − 0.193i)21-s + (−0.0408 + 0.152i)23-s + (0.203 + 0.203i)27-s + (1.22 − 0.706i)29-s + (0.390 − 0.390i)31-s + (−0.0910 + 0.157i)33-s + (−0.882 + 0.509i)37-s + (0.107 + 0.0978i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.492 + 0.870i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.492 + 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.586327646\)
\(L(\frac12)\) \(\approx\) \(1.586327646\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 + (3.20 + 1.65i)T \)
good3 \( 1 + (0.243 + 0.0651i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-4.30 - 2.48i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.07 + 4.01i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.65 + 6.16i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.01 + 0.270i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.195 - 0.730i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-6.59 + 3.80i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.17 + 2.17i)T - 31iT^{2} \)
37 \( 1 + (5.36 - 3.09i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.57 + 0.689i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.90 + 0.779i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 1.62iT - 47T^{2} \)
53 \( 1 + (-5.32 + 5.32i)T - 53iT^{2} \)
59 \( 1 + (-2.52 - 9.43i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-5.39 + 9.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.04 - 5.27i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.15 + 4.31i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 5.98T + 73T^{2} \)
79 \( 1 + 14.0iT - 79T^{2} \)
83 \( 1 + 8.32iT - 83T^{2} \)
89 \( 1 + (1.99 + 0.534i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (4.17 - 7.23i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.301234932102673976349302392226, −8.611520157722119140999104520049, −8.133693904288958623427542835219, −7.10432994598162681438218333756, −5.98350873768194037275322916309, −5.31178976217734409185346315253, −4.64681102096478717927349469737, −3.12331628053422296047441687222, −2.31975721439090855577255724327, −0.70847116977019333008913140817, 1.44314454128430276559111683130, 2.34537224275458173039944546521, 4.01924583741293486601933867880, 4.68149844496981630143338403980, 5.34587389617774014807771209391, 6.68204579436431847985569758303, 7.36346295769396862082330820957, 8.195764451680824197651278970085, 8.753707591654694659980143743813, 10.03122961212978854480009960400

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