↑

Properties

Label 2-650-13.4-c1-0-20
Degree $2$
Conductor $650$
Sign $-0.856 + 0.515i$
Analytic cond. $5.19027$
Root an. cond. $2.27821$
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  + (−0.866 − 0.5i)2-s + (1.66 − 2.88i)3-s + (0.499 + 0.866i)4-s + (−2.88 + 1.66i)6-s + (1.24 − 0.719i)7-s − 0.999i·8-s + (−4.05 − 7.01i)9-s + (3.49 + 2.01i)11-s + 3.33·12-s + (−3.13 − 1.78i)13-s − 1.43·14-s + (−0.5 + 0.866i)16-s + (−1.15 − 1.99i)17-s + 8.10i·18-s + (−0.346 + 0.199i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.961 − 1.66i)3-s + (0.249 + 0.433i)4-s + (−1.17 + 0.680i)6-s + (0.471 − 0.272i)7-s − 0.353i·8-s + (−1.35 − 2.33i)9-s + (1.05 + 0.608i)11-s + 0.961·12-s + (−0.868 − 0.495i)13-s − 0.384·14-s + (−0.125 + 0.216i)16-s + (−0.279 − 0.484i)17-s + 1.91i·18-s + (−0.0794 + 0.0458i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.856 + 0.515i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ -0.856 + 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.402085 - 1.44746i\)
\(L(\frac12)\) \(\approx\) \(0.402085 - 1.44746i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (3.13 + 1.78i)T \)
good3 \( 1 + (-1.66 + 2.88i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-1.24 + 0.719i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.49 - 2.01i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.15 + 1.99i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.346 - 0.199i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.780 - 1.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.93 + 6.81i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.10iT - 31T^{2} \)
37 \( 1 + (-5.40 - 3.11i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.659 + 0.380i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.50 - 9.53i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.84iT - 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 + (3 - 1.73i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.84 - 4.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.00 - 5.20i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.19 + 3i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 13.4iT - 73T^{2} \)
79 \( 1 - 0.931T + 79T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 + (0.840 + 0.485i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.6 + 6.15i)T + (48.5 - 84.0i)T^{2} \)
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

−9.822415919006758589705018372850, −9.296029057015670049532572080196, −8.219088311321869400740740289323, −7.72559843443547647020188246316, −6.98133904902296607329253938299, −6.16369695786025063708323013586, −4.32009640056024267301893065646, −2.92185926858725771597366357339, −2.03391840446011864138283540469, −0.916409003067604617800974685706, 2.10553355259919227015654739054, 3.37920893419240691638610187855, 4.43756462154625976499907280128, 5.23183256211632479761528593921, 6.50336594851921932372258972963, 7.81060840418299163109564913152, 8.624090173491049028955569657646, 9.121820714467985695277558521765, 9.775523686937522801908911937033, 10.73009417161360334705653599683

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