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Properties

Label 2-650-13.10-c1-0-18
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

\(L(1)\) \(\approx\) \(0.994379 - 1.57484i\)
\(L(\frac12)\) \(\approx\) \(0.994379 - 1.57484i\)
\(L(\frac{3}{2})\) 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

−10.46778373019766291633543683855, −9.580321529026548180129033871091, −8.424952993772464433538711253887, −7.47935174661337260291636371710, −6.49597798507149485172207961101, −5.79988213477781207861883975325, −4.77331722766758311865224030317, −3.60021481896160018344476288086, −2.26974118419329122282582743459, −0.922298214810271599555416774422, 1.93233779481176857430971770910, 3.61665219413775578709361311852, 4.54437666918776850974856906610, 5.09973754359526540455779844078, 6.23883685557796824590593578548, 7.23607410179045581832527263172, 7.980291532375094466258655515576, 9.330659483963410129573322578924, 9.911417120651321038638891787784, 11.00564267165457999891970534043

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