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Properties

Label 2-650-65.2-c1-0-15
Degree $2$
Conductor $650$
Sign $-0.444 + 0.895i$
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 + (1.01 + 0.272i)3-s + (−0.499 + 0.866i)4-s + (−0.272 − 1.01i)6-s + (−0.524 − 0.303i)7-s + 0.999·8-s + (−1.63 − 0.944i)9-s + (1.67 − 6.24i)11-s + (−0.745 + 0.745i)12-s + (−3.55 + 0.572i)13-s + 0.606i·14-s + (−0.5 − 0.866i)16-s + (0.267 + 0.996i)17-s + 1.88i·18-s + (0.896 − 0.240i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.587 + 0.157i)3-s + (−0.249 + 0.433i)4-s + (−0.111 − 0.415i)6-s + (−0.198 − 0.114i)7-s + 0.353·8-s + (−0.545 − 0.314i)9-s + (0.504 − 1.88i)11-s + (−0.215 + 0.215i)12-s + (−0.987 + 0.158i)13-s + 0.162i·14-s + (−0.125 − 0.216i)16-s + (0.0647 + 0.241i)17-s + 0.445i·18-s + (0.205 − 0.0551i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.444 + 0.895i$
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: \((2,\ 650,\ (\ :1/2),\ -0.444 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.601390 - 0.969816i\)
\(L(\frac12)\) \(\approx\) \(0.601390 - 0.969816i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.55 - 0.572i)T \)
good3 \( 1 + (-1.01 - 0.272i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.524 + 0.303i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.67 + 6.24i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.267 - 0.996i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.896 + 0.240i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.31 + 4.90i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.65 + 1.53i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.06 + 7.06i)T - 31iT^{2} \)
37 \( 1 + (0.0372 - 0.0214i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.75 - 0.471i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (6.78 - 1.81i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 - 7.31iT - 47T^{2} \)
53 \( 1 + (-3.80 + 3.80i)T - 53iT^{2} \)
59 \( 1 + (1.52 + 5.68i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.28 - 2.21i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.28 - 5.69i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.71 - 10.1i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 2.04T + 73T^{2} \)
79 \( 1 - 4.09iT - 79T^{2} \)
83 \( 1 - 7.40iT - 83T^{2} \)
89 \( 1 + (14.6 + 3.91i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-8.80 + 15.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.10195285144228253817554288348, −9.469883299697114706470980542809, −8.489100783591493068966845483811, −8.173987813272685575016436685092, −6.72564488623917513992764687741, −5.79691521323199526242315529519, −4.36606396526152303415887995322, −3.30017404241711820895254420507, −2.56686346745853954549385517983, −0.64369963136564601401032463651, 1.79197391499946903380810751165, 3.02168733563119695824954787989, 4.58351005991489998382710647017, 5.34137437610281156195506823443, 6.72592440924782611889788076068, 7.33384851771457640626475314297, 8.118434728683662550751089589540, 9.136539593115415823964082092136, 9.707905245438556284702031378220, 10.49245338457614352545588482899

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