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Properties

Label 2-130-13.10-c1-0-3
Degree $2$
Conductor $130$
Sign $0.964 - 0.265i$
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 + (1.36 + 2.36i)3-s + (0.499 − 0.866i)4-s − i·5-s + (2.36 + 1.36i)6-s + (−2.59 − 1.5i)7-s − 0.999i·8-s + (−2.23 + 3.86i)9-s + (−0.5 − 0.866i)10-s + (−2.59 + 1.5i)11-s + 2.73·12-s + (3.5 + 0.866i)13-s − 3·14-s + (2.36 − 1.36i)15-s + (−0.5 − 0.866i)16-s + (1.09 − 1.90i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.788 + 1.36i)3-s + (0.249 − 0.433i)4-s − 0.447i·5-s + (0.965 + 0.557i)6-s + (−0.981 − 0.566i)7-s − 0.353i·8-s + (−0.744 + 1.28i)9-s + (−0.158 − 0.273i)10-s + (−0.783 + 0.452i)11-s + 0.788·12-s + (0.970 + 0.240i)13-s − 0.801·14-s + (0.610 − 0.352i)15-s + (−0.125 − 0.216i)16-s + (0.266 − 0.461i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ 0.964 - 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63156 + 0.220123i\)
\(L(\frac12)\) \(\approx\) \(1.63156 + 0.220123i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
13 \( 1 + (-3.5 - 0.866i)T \)
good3 \( 1 + (-1.36 - 2.36i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.59 + 1.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.09 + 1.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.59 + 3.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.26 + 2.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.73 - 8.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.26iT - 31T^{2} \)
37 \( 1 + (-9.69 + 5.59i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (9 - 5.19i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 + 6.46T + 53T^{2} \)
59 \( 1 + (-9 - 5.19i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.09 + 3.63i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.19 - 3i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 5.66iT - 73T^{2} \)
79 \( 1 - 6.19T + 79T^{2} \)
83 \( 1 - 2.19iT - 83T^{2} \)
89 \( 1 + (14.8 - 8.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.0 - 7.56i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.35266138746201907577406168461, −12.66266728826613885530770339697, −11.01120198299239680712090565881, −10.25634559841841432570749836031, −9.413540063681346633807313250126, −8.378209864678558874944246394307, −6.60775075236474066458397663898, −4.95107123783892681423417044235, −4.02166217655859565416862422534, −2.87863066665948351263053362496, 2.39557014566654468148346202004, 3.53290408299305136501776025453, 5.99333068932491202487180354626, 6.50324869531993387509419249081, 7.921876494092038646709983613602, 8.494171274888494869114445454004, 10.15754233316880493629831271641, 11.65052394323239487029667841000, 12.75055297869982957570873778678, 13.22260046351494205593445704690

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