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Properties

Label 2-650-65.37-c1-0-17
Degree $2$
Conductor $650$
Sign $-0.939 + 0.342i$
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 + (−1.09 − 0.294i)3-s + (0.499 − 0.866i)4-s + (1.09 − 0.294i)6-s + (2.54 − 4.40i)7-s + 0.999i·8-s + (−1.47 − 0.854i)9-s + (−2.27 − 0.609i)11-s + (−0.803 + 0.803i)12-s + (−3.54 − 0.653i)13-s + 5.08i·14-s + (−0.5 − 0.866i)16-s + (0.0601 + 0.224i)17-s + 1.70·18-s + (1.53 + 5.71i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.633 − 0.169i)3-s + (0.249 − 0.433i)4-s + (0.448 − 0.120i)6-s + (0.961 − 1.66i)7-s + 0.353i·8-s + (−0.493 − 0.284i)9-s + (−0.685 − 0.183i)11-s + (−0.231 + 0.231i)12-s + (−0.983 − 0.181i)13-s + 1.35i·14-s + (−0.125 − 0.216i)16-s + (0.0145 + 0.0544i)17-s + 0.402·18-s + (0.351 + 1.31i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0542787 - 0.306960i\)
\(L(\frac12)\) \(\approx\) \(0.0542787 - 0.306960i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.54 + 0.653i)T \)
good3 \( 1 + (1.09 + 0.294i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-2.54 + 4.40i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.27 + 0.609i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.0601 - 0.224i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.53 - 5.71i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.674 - 2.51i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.64 - 1.52i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.45 + 4.45i)T + 31iT^{2} \)
37 \( 1 + (2.49 + 4.31i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.412 - 1.53i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.66 - 0.447i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 + (5.79 - 5.79i)T - 53iT^{2} \)
59 \( 1 + (2.68 - 0.720i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.23 + 7.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.150 - 0.0866i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.17 + 1.11i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + 7.51iT - 73T^{2} \)
79 \( 1 - 5.17iT - 79T^{2} \)
83 \( 1 - 10.4T + 83T^{2} \)
89 \( 1 + (-1.49 + 5.59i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-7.02 - 4.05i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.27058273699329392257004160943, −9.430615976513263840028970122677, −7.995008060268446715229801747507, −7.71808010727264829210805679104, −6.78104767369666507467219436821, −5.64239051375283223029398371040, −4.88363399651925718654090291445, −3.53586096449382512429720887694, −1.63945699087803674788010051372, −0.21382011072114182108978444474, 2.06000749279195726190727307282, 2.84092410066524580278693924012, 5.00209795849935858880149515756, 5.14196868209246119830969772146, 6.46605955890926345867651637610, 7.67305739000469605666099103095, 8.475884353214773168234702113780, 9.160257466920681167882654671115, 10.12043223724868710451209618882, 11.11287477232795605796382691363

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