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Properties

Label 2-650-13.10-c1-0-16
Degree $2$
Conductor $650$
Sign $0.201 + 0.979i$
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 + (0.918 + 1.59i)3-s + (0.499 − 0.866i)4-s + (−1.59 − 0.918i)6-s + (−4.15 − 2.39i)7-s + 0.999i·8-s + (−0.188 + 0.326i)9-s + (1.43 − 0.825i)11-s + 1.83·12-s + (−3.43 − 1.09i)13-s + 4.79·14-s + (−0.5 − 0.866i)16-s + (−0.0402 + 0.0697i)17-s − 0.376i·18-s + (−3.67 − 2.12i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.530 + 0.918i)3-s + (0.249 − 0.433i)4-s + (−0.649 − 0.375i)6-s + (−1.57 − 0.906i)7-s + 0.353i·8-s + (−0.0628 + 0.108i)9-s + (0.431 − 0.248i)11-s + 0.530·12-s + (−0.953 − 0.302i)13-s + 1.28·14-s + (−0.125 − 0.216i)16-s + (−0.00976 + 0.0169i)17-s − 0.0888i·18-s + (−0.843 − 0.487i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.201 + 0.979i$
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: \((2,\ 650,\ (\ :1/2),\ 0.201 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.435151 - 0.354620i\)
\(L(\frac12)\) \(\approx\) \(0.435151 - 0.354620i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.43 + 1.09i)T \)
good3 \( 1 + (-0.918 - 1.59i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (4.15 + 2.39i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.43 + 0.825i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.0402 - 0.0697i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.67 + 2.12i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.94 + 5.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.21 + 3.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.06iT - 31T^{2} \)
37 \( 1 + (-1.57 + 0.908i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.87 - 3.39i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.24 - 7.35i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.448iT - 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + (1.82 + 1.05i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.56 + 2.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.46 - 2i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.36 + 3.67i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.5iT - 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 + 12.4iT - 83T^{2} \)
89 \( 1 + (7.23 - 4.17i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.02 - 2.90i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.07974115587190221339747537750, −9.600938480493482026765206397596, −8.875437605026425322820692599731, −7.82096884045716721600677670275, −6.76915628784202634086401903742, −6.19661296950493502950780959354, −4.59996714543433085983961382684, −3.74454693674741203480016772859, −2.66276593147540981566056160033, −0.32487661436447590860013068401, 1.80612300685439042158602165239, 2.67382114180383756675440720650, 3.76435087860111446314530509161, 5.51246135271893574448401982162, 6.72825307972234006968543672266, 7.11848569818155909240997415456, 8.286016690628472482205843008192, 9.020062213416595862502144724171, 9.747303382470178249530705613236, 10.47034740193202024807446582121

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