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Properties

Label 2-130-65.28-c1-0-5
Degree $2$
Conductor $130$
Sign $-0.204 + 0.978i$
Analytic cond. $1.03805$
Root an. cond. $1.01884$
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  + (−0.866 + 0.5i)2-s + (0.814 − 3.04i)3-s + (0.499 − 0.866i)4-s + (−2.22 − 0.208i)5-s + (0.814 + 3.04i)6-s + (0.402 − 0.696i)7-s + 0.999i·8-s + (−5.98 − 3.45i)9-s + (2.03 − 0.932i)10-s + (−0.778 + 2.90i)11-s + (−2.22 − 2.22i)12-s + (0.206 − 3.59i)13-s + 0.804i·14-s + (−2.44 + 6.60i)15-s + (−0.5 − 0.866i)16-s + (6.99 − 1.87i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.470 − 1.75i)3-s + (0.249 − 0.433i)4-s + (−0.995 − 0.0931i)5-s + (0.332 + 1.24i)6-s + (0.152 − 0.263i)7-s + 0.353i·8-s + (−1.99 − 1.15i)9-s + (0.642 − 0.294i)10-s + (−0.234 + 0.876i)11-s + (−0.642 − 0.642i)12-s + (0.0571 − 0.998i)13-s + 0.215i·14-s + (−0.631 + 1.70i)15-s + (−0.125 − 0.216i)16-s + (1.69 − 0.454i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $-0.204 + 0.978i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ -0.204 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.495085 - 0.609361i\)
\(L(\frac12)\) \(\approx\) \(0.495085 - 0.609361i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (2.22 + 0.208i)T \)
13 \( 1 + (-0.206 + 3.59i)T \)
good3 \( 1 + (-0.814 + 3.04i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-0.402 + 0.696i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.778 - 2.90i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-6.99 + 1.87i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-4.51 + 1.21i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.422 - 0.113i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.58 - 2.06i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.536 - 0.536i)T - 31iT^{2} \)
37 \( 1 + (0.482 + 0.835i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.63 - 0.437i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.64 - 6.14i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 1.72T + 47T^{2} \)
53 \( 1 + (5.01 + 5.01i)T + 53iT^{2} \)
59 \( 1 + (0.0422 + 0.157i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.11 + 1.93i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.82 + 2.20i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.63 - 9.83i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 - 7.75iT - 79T^{2} \)
83 \( 1 + 15.6T + 83T^{2} \)
89 \( 1 + (-5.91 - 1.58i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (11.6 + 6.70i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.82703674231732484571151054494, −12.21996778639742248156125592150, −11.18703951799765987045203816490, −9.631592738551090329584692131579, −8.232036641847441072056254095583, −7.60254914359114372814537069198, −7.10636013373435566592995294194, −5.47813214442446612496689547634, −3.05061419569195185343293834072, −1.05326644270254489554451018446, 3.16793388237779430655519233887, 3.99381184012509492380932240858, 5.48959146359871244505335199626, 7.69464623653160892542532596312, 8.581790556342156993043725983889, 9.463693976157284549147439824083, 10.43684476310967409079384164074, 11.30255756336390179115109682859, 12.05777205854869509317821257732, 13.94598447331454961430182046483

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