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Properties

Label 2-650-65.7-c1-0-10
Degree $2$
Conductor $650$
Sign $0.999 + 0.0421i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.316 + 1.18i)3-s + (0.499 + 0.866i)4-s + (0.316 − 1.18i)6-s + (0.401 + 0.695i)7-s − 0.999i·8-s + (1.30 − 0.752i)9-s + (−0.707 − 2.64i)11-s + (−0.864 + 0.864i)12-s + (1.91 − 3.05i)13-s − 0.803i·14-s + (−0.5 + 0.866i)16-s + (3.64 + 0.975i)17-s − 1.50·18-s + (−2.03 − 0.544i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.182 + 0.682i)3-s + (0.249 + 0.433i)4-s + (0.129 − 0.482i)6-s + (0.151 + 0.262i)7-s − 0.353i·8-s + (0.434 − 0.250i)9-s + (−0.213 − 0.796i)11-s + (−0.249 + 0.249i)12-s + (0.532 − 0.846i)13-s − 0.214i·14-s + (−0.125 + 0.216i)16-s + (0.883 + 0.236i)17-s − 0.354·18-s + (−0.466 − 0.124i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.999 + 0.0421i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.999 + 0.0421i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30015 - 0.0273899i\)
\(L(\frac12)\) \(\approx\) \(1.30015 - 0.0273899i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.91 + 3.05i)T \)
good3 \( 1 + (-0.316 - 1.18i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.401 - 0.695i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.707 + 2.64i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-3.64 - 0.975i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.03 + 0.544i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.71 + 1.26i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.08 + 1.20i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.16 - 4.16i)T + 31iT^{2} \)
37 \( 1 + (3.07 - 5.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.97 + 1.33i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.78 - 6.64i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 4.44T + 47T^{2} \)
53 \( 1 + (-9.13 + 9.13i)T - 53iT^{2} \)
59 \( 1 + (2.20 - 8.21i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-7.35 - 12.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.44 + 4.30i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.83 + 14.3i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 1.70iT - 73T^{2} \)
79 \( 1 - 1.85iT - 79T^{2} \)
83 \( 1 - 1.38T + 83T^{2} \)
89 \( 1 + (1.87 - 0.501i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.41 - 1.39i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.41705387661569593747750428388, −9.837142024543322827695859028154, −8.746107689062435729992313397520, −8.333279005202978628923740397635, −7.17690925557680327039408617122, −6.04543813584767318181629150246, −4.95296368017450705732797085736, −3.68495145000788140722198837394, −2.89029780751251790613212159136, −1.10119929498304661085412075809, 1.23398440460636313529542128422, 2.32827442909622300279710981800, 4.05331616780267970680813162184, 5.19610698333744993962916909977, 6.42151748950163129695878135287, 7.22951726479180551499710275862, 7.73916613225121862102382561077, 8.740268607475063063475155020259, 9.620188251619181122592927544605, 10.42657516636781243245615636943

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