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Properties

Label 2-650-25.6-c1-0-20
Degree $2$
Conductor $650$
Sign $0.941 - 0.338i$
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  + (0.309 + 0.951i)2-s + (2.22 − 1.61i)3-s + (−0.809 + 0.587i)4-s + (0.656 + 2.13i)5-s + (2.22 + 1.61i)6-s + 1.40·7-s + (−0.809 − 0.587i)8-s + (1.41 − 4.36i)9-s + (−1.82 + 1.28i)10-s + (−1.79 − 5.53i)11-s + (−0.851 + 2.61i)12-s + (0.309 − 0.951i)13-s + (0.432 + 1.33i)14-s + (4.92 + 3.69i)15-s + (0.309 − 0.951i)16-s + (5.84 + 4.24i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (1.28 − 0.934i)3-s + (−0.404 + 0.293i)4-s + (0.293 + 0.955i)5-s + (0.909 + 0.660i)6-s + 0.529·7-s + (−0.286 − 0.207i)8-s + (0.472 − 1.45i)9-s + (−0.578 + 0.406i)10-s + (−0.542 − 1.66i)11-s + (−0.245 + 0.756i)12-s + (0.0857 − 0.263i)13-s + (0.115 + 0.356i)14-s + (1.27 + 0.955i)15-s + (0.0772 − 0.237i)16-s + (1.41 + 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.338i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.941 - 0.338i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.941 - 0.338i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.60830 + 0.454311i\)
\(L(\frac12)\) \(\approx\) \(2.60830 + 0.454311i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.656 - 2.13i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (-2.22 + 1.61i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 - 1.40T + 7T^{2} \)
11 \( 1 + (1.79 + 5.53i)T + (-8.89 + 6.46i)T^{2} \)
17 \( 1 + (-5.84 - 4.24i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-4.10 - 2.97i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.179 + 0.552i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (4.80 - 3.49i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (4.07 + 2.96i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.739 - 2.27i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.54 + 4.75i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.43T + 43T^{2} \)
47 \( 1 + (8.32 - 6.04i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (5.29 - 3.84i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.28 + 10.1i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.33 - 7.17i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (9.12 + 6.62i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (2.57 - 1.87i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.728 - 2.24i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-8.03 + 5.83i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (11.0 + 8.05i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.68 + 8.25i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-4.62 + 3.36i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.54396098538986860409012836061, −9.478053453461268025924831184650, −8.437294990684513415267623783213, −7.85802898531931491781279095942, −7.39411928873043155080664525827, −6.12728577433464468867321190889, −5.55927008377746127832744367859, −3.48698627018386290476373861208, −3.13439078166623079721051284682, −1.58443322207975368504055158051, 1.65448136406194516611614881517, 2.72589287677672813252657287303, 3.88913209904037427109994760315, 4.86273952016806255508864719218, 5.25888806192559910923916451834, 7.39156756495444847465016574182, 8.086373283636728220679338060348, 9.160832352347897536673807227481, 9.661018354812091648516626789336, 10.05885021007436554024109994320

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