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Properties

Label 2-650-65.32-c1-0-10
Degree $2$
Conductor $650$
Sign $0.997 - 0.0668i$
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  + (0.5 − 0.866i)2-s + (0.617 + 2.30i)3-s + (−0.499 − 0.866i)4-s + (2.30 + 0.617i)6-s + (2.54 − 1.46i)7-s − 0.999·8-s + (−2.33 + 1.34i)9-s + (1.03 − 0.278i)11-s + (1.68 − 1.68i)12-s + (2.75 − 2.32i)13-s − 2.93i·14-s + (−0.5 + 0.866i)16-s + (1.41 + 0.378i)17-s + 2.69i·18-s + (−0.564 + 2.10i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.356 + 1.33i)3-s + (−0.249 − 0.433i)4-s + (0.940 + 0.252i)6-s + (0.961 − 0.555i)7-s − 0.353·8-s + (−0.777 + 0.448i)9-s + (0.313 − 0.0839i)11-s + (0.487 − 0.487i)12-s + (0.764 − 0.644i)13-s − 0.785i·14-s + (−0.125 + 0.216i)16-s + (0.342 + 0.0918i)17-s + 0.634i·18-s + (−0.129 + 0.483i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.997 - 0.0668i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.997 - 0.0668i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23128 + 0.0746336i\)
\(L(\frac12)\) \(\approx\) \(2.23128 + 0.0746336i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
13 \( 1 + (-2.75 + 2.32i)T \)
good3 \( 1 + (-0.617 - 2.30i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-2.54 + 1.46i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.03 + 0.278i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.41 - 0.378i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.564 - 2.10i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.19 + 0.320i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-6.21 - 3.58i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.92 - 5.92i)T - 31iT^{2} \)
37 \( 1 + (9.51 + 5.49i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.01 - 3.78i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.88 - 7.03i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 13.0iT - 47T^{2} \)
53 \( 1 + (-8.91 + 8.91i)T - 53iT^{2} \)
59 \( 1 + (-2.01 - 0.540i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.289 - 0.500i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.47 - 7.74i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.6 + 3.39i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 5.77T + 73T^{2} \)
79 \( 1 + 6.64iT - 79T^{2} \)
83 \( 1 + 0.332iT - 83T^{2} \)
89 \( 1 + (1.94 + 7.25i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (5.95 + 10.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.49398958298621397391606307845, −10.07675715596413648847353712512, −8.843328817954734487636051581727, −8.396094040504539450373431371758, −7.02617855284425249124657950426, −5.55791923237876351919553016065, −4.82082149293541806079383187185, −3.87697206013592533834281938422, −3.21958090953741442329941180293, −1.45800056122657036064998824909, 1.41745194495979884294346547537, 2.55612397558494399032673555305, 4.07425847516447265429814095210, 5.24651814484669832308092808030, 6.25353513063061810534307750122, 7.00608363148550125659686831665, 7.82964292645945500744416038901, 8.530239690885538535285994215468, 9.205293476027498272197068805935, 10.74774829014395427207929092549

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