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Properties

Label 2-130-65.28-c1-0-1
Degree $2$
Conductor $130$
Sign $0.536 - 0.844i$
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 + (−0.640 + 2.38i)3-s + (0.499 − 0.866i)4-s + (−0.606 + 2.15i)5-s + (0.640 + 2.38i)6-s + (−0.551 + 0.954i)7-s − 0.999i·8-s + (−2.70 − 1.56i)9-s + (0.551 + 2.16i)10-s + (1.26 − 4.73i)11-s + (1.74 + 1.74i)12-s + (3.50 + 0.841i)13-s + 1.10i·14-s + (−4.75 − 2.82i)15-s + (−0.5 − 0.866i)16-s + (4.62 − 1.23i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.369 + 1.37i)3-s + (0.249 − 0.433i)4-s + (−0.271 + 0.962i)5-s + (0.261 + 0.975i)6-s + (−0.208 + 0.360i)7-s − 0.353i·8-s + (−0.901 − 0.520i)9-s + (0.174 + 0.685i)10-s + (0.382 − 1.42i)11-s + (0.505 + 0.505i)12-s + (0.972 + 0.233i)13-s + 0.294i·14-s + (−1.22 − 0.730i)15-s + (−0.125 − 0.216i)16-s + (1.12 − 0.300i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.536 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.536 - 0.844i$
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: \((2,\ 130,\ (\ :1/2),\ 0.536 - 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12847 + 0.620020i\)
\(L(\frac12)\) \(\approx\) \(1.12847 + 0.620020i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.606 - 2.15i)T \)
13 \( 1 + (-3.50 - 0.841i)T \)
good3 \( 1 + (0.640 - 2.38i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (0.551 - 0.954i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.26 + 4.73i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-4.62 + 1.23i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.778 - 0.208i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (5.22 + 1.39i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-6.12 + 3.53i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.44 - 2.44i)T - 31iT^{2} \)
37 \( 1 + (0.706 + 1.22i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (11.3 + 3.03i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.972 - 3.63i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 0.538T + 47T^{2} \)
53 \( 1 + (-3.71 - 3.71i)T + 53iT^{2} \)
59 \( 1 + (3.24 + 12.1i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.395 - 0.685i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.6 + 6.13i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.147 + 0.549i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 - 3.94iT - 73T^{2} \)
79 \( 1 - 9.74iT - 79T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 + (-3.78 - 1.01i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (12.1 + 7.00i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.83999552567688030805462258090, −12.10864648501003662994187021192, −11.28512636490336417057068100925, −10.59578318364139915633724342867, −9.725712431505338569617619614327, −8.367308169427029681706553702787, −6.41744778483880655588336493810, −5.56794465030110417846265621471, −3.97513260465008365214853955811, −3.19673721399594053573022496804, 1.53074757636986622497649330778, 3.97124641265687415784435722979, 5.44039122792061239336027539080, 6.59891870449773402728593509387, 7.54121321734366785599179210092, 8.460348456030377224848047783886, 10.09199194699118476356418840798, 11.84063419765833852655904878538, 12.25098090998504955575721153256, 13.07820853526281459711701984458

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