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Properties

Label 2-650-13.10-c1-0-5
Degree $2$
Conductor $650$
Sign $0.429 - 0.902i$
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 + (0.747 + 1.29i)3-s + (0.499 − 0.866i)4-s + (−1.29 − 0.747i)6-s + (−1.21 − 0.700i)7-s + 0.999i·8-s + (0.381 − 0.661i)9-s + (1.99 − 1.15i)11-s + 1.49·12-s + (2.08 + 2.94i)13-s + 1.40·14-s + (−0.5 − 0.866i)16-s + (0.286 − 0.496i)17-s + 0.763i·18-s + (2.38 + 1.37i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.431 + 0.747i)3-s + (0.249 − 0.433i)4-s + (−0.528 − 0.305i)6-s + (−0.458 − 0.264i)7-s + 0.353i·8-s + (0.127 − 0.220i)9-s + (0.602 − 0.347i)11-s + 0.431·12-s + (0.577 + 0.816i)13-s + 0.374·14-s + (−0.125 − 0.216i)16-s + (0.0695 − 0.120i)17-s + 0.179i·18-s + (0.547 + 0.316i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.429 - 0.902i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.429 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13059 + 0.713871i\)
\(L(\frac12)\) \(\approx\) \(1.13059 + 0.713871i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.08 - 2.94i)T \)
good3 \( 1 + (-0.747 - 1.29i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.21 + 0.700i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.99 + 1.15i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.286 + 0.496i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.38 - 1.37i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.72 - 6.45i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.65 - 4.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.52iT - 31T^{2} \)
37 \( 1 + (2.24 - 1.29i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.21 - 0.701i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.570 - 0.988i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.78iT - 47T^{2} \)
53 \( 1 - 0.889T + 53T^{2} \)
59 \( 1 + (-2.04 - 1.18i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.67 - 8.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.87 - 2.23i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.77 - 4.48i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.79iT - 73T^{2} \)
79 \( 1 - 15.9T + 79T^{2} \)
83 \( 1 - 8.36iT - 83T^{2} \)
89 \( 1 + (1.93 - 1.11i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.9 + 8.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.44625219434155332899789774138, −9.561907463055822895915349473011, −9.181195237080019282614145825481, −8.330216199058749507778650273567, −7.12383444064472255971912208728, −6.47426805298124861957546624514, −5.29206981241289988050387416695, −3.99992020798941671894920134829, −3.21298436863247984441573885616, −1.31114779702372798767453288138, 1.04526040114727832531583130824, 2.37466633995313363767191870586, 3.34140861897487045845061249806, 4.78843874318374540039808073705, 6.24992449174196798654653977427, 6.99103748516573913909956833428, 7.924579398074231049018887521303, 8.612069988970483195645511769999, 9.452556564782451342881458856858, 10.38104942377473310267564509641

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