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Properties

Label 2-650-13.3-c1-0-2
Degree $2$
Conductor $650$
Sign $0.859 + 0.511i$
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 + (−1.39 − 2.41i)3-s + (−0.499 + 0.866i)4-s + (−1.39 + 2.41i)6-s + (−0.895 + 1.55i)7-s + 0.999·8-s + (−2.39 + 4.14i)9-s + (1.89 + 3.28i)11-s + 2.79·12-s + (3.5 − 0.866i)13-s + 1.79·14-s + (−0.5 − 0.866i)16-s + (−0.395 + 0.685i)17-s + 4.79·18-s + (−2.5 + 4.33i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.805 − 1.39i)3-s + (−0.249 + 0.433i)4-s + (−0.569 + 0.986i)6-s + (−0.338 + 0.586i)7-s + 0.353·8-s + (−0.798 + 1.38i)9-s + (0.571 + 0.989i)11-s + 0.805·12-s + (0.970 − 0.240i)13-s + 0.478·14-s + (−0.125 − 0.216i)16-s + (−0.0959 + 0.166i)17-s + 1.12·18-s + (−0.573 + 0.993i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.773115 - 0.212477i\)
\(L(\frac12)\) \(\approx\) \(0.773115 - 0.212477i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 + 0.866i)T \)
good3 \( 1 + (1.39 + 2.41i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.895 - 1.55i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.89 - 3.28i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.395 - 0.685i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.29 - 3.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.89 - 3.28i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.58T + 31T^{2} \)
37 \( 1 + (3.18 + 5.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.08 + 10.5i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.47 - 9.48i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.791T + 47T^{2} \)
53 \( 1 + 4.58T + 53T^{2} \)
59 \( 1 + (-2.29 + 3.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.29 + 7.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.79 - 6.56i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.16T + 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 - 6.16T + 83T^{2} \)
89 \( 1 + (0.313 + 0.542i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.60 - 7.97i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.66145193590802923465922085317, −9.709358749536009413593970577245, −8.653870325389927456284616336866, −7.88246157812327075378120619852, −6.82133486722961748545813919768, −6.23378641864434210073030402591, −5.14845312164131762986691983518, −3.64931479427102499146487111535, −2.15097070868194877806978788464, −1.21024399847144461108814141273, 0.64505215125806090537328673785, 3.32836148464097840776418341979, 4.33306039641488321028035204226, 5.07888872865135585223955183218, 6.39978575987509624126144693386, 6.55138584564606777982085187449, 8.333867336196578039554479996354, 8.895962007994506389438605165525, 9.882385898839332687477026887456, 10.48645303801781960179820124972

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