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Properties

Label 2-650-65.2-c1-0-17
Degree $2$
Conductor $650$
Sign $0.191 + 0.981i$
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 + (0.278 + 0.0746i)3-s + (−0.499 + 0.866i)4-s + (0.0746 + 0.278i)6-s + (−4.15 − 2.39i)7-s − 0.999·8-s + (−2.52 − 1.45i)9-s + (−0.105 + 0.395i)11-s + (−0.203 + 0.203i)12-s + (2.47 − 2.61i)13-s − 4.79i·14-s + (−0.5 − 0.866i)16-s + (−0.999 − 3.73i)17-s − 2.91i·18-s + (5.05 − 1.35i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.160 + 0.0430i)3-s + (−0.249 + 0.433i)4-s + (0.0304 + 0.113i)6-s + (−1.56 − 0.905i)7-s − 0.353·8-s + (−0.842 − 0.486i)9-s + (−0.0319 + 0.119i)11-s + (−0.0588 + 0.0588i)12-s + (0.687 − 0.726i)13-s − 1.28i·14-s + (−0.125 − 0.216i)16-s + (−0.242 − 0.904i)17-s − 0.687i·18-s + (1.16 − 0.311i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.191 + 0.981i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.191 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.679025 - 0.559303i\)
\(L(\frac12)\) \(\approx\) \(0.679025 - 0.559303i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.47 + 2.61i)T \)
good3 \( 1 + (-0.278 - 0.0746i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (4.15 + 2.39i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.105 - 0.395i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.999 + 3.73i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-5.05 + 1.35i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.60 + 5.97i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (8.13 - 4.69i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.0701 + 0.0701i)T - 31iT^{2} \)
37 \( 1 + (9.07 - 5.23i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.48 - 1.46i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (7.42 - 1.99i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 6.95iT - 47T^{2} \)
53 \( 1 + (3.07 - 3.07i)T - 53iT^{2} \)
59 \( 1 + (0.564 + 2.10i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.13 + 3.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.986 + 1.70i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.91 + 10.8i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 0.776T + 73T^{2} \)
79 \( 1 + 1.19iT - 79T^{2} \)
83 \( 1 + 11.3iT - 83T^{2} \)
89 \( 1 + (3.47 + 0.931i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-6.51 + 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.23329005956937307840956115106, −9.366621199095562573500244766633, −8.685046253128207211038366834966, −7.47091031392166716053359413409, −6.77190589013837156872711960017, −5.99385263966694402816170455613, −4.93683390533176760789601064302, −3.46648162586341988856010773583, −3.12480466314687537899488802600, −0.39569283310829816424964694910, 1.94231610980297535279475997899, 3.13596414684201759744445861774, 3.80237019810686282491156481225, 5.56925514692140970804413941175, 5.86981125912721670034526491652, 7.08760917441680299501097122897, 8.412797240508871416571926766876, 9.231438066645538298713592033624, 9.732041655092254422269740158106, 10.91671216701489436011318268811

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