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Properties

Label 2-130-65.37-c1-0-5
Degree $2$
Conductor $130$
Sign $0.886 + 0.462i$
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 + (1.09 + 0.294i)3-s + (0.499 − 0.866i)4-s + (0.803 − 2.08i)5-s + (1.09 − 0.294i)6-s + (−2.54 + 4.40i)7-s − 0.999i·8-s + (−1.47 − 0.854i)9-s + (−0.347 − 2.20i)10-s + (−2.27 − 0.609i)11-s + (0.803 − 0.803i)12-s + (3.54 + 0.653i)13-s + 5.08i·14-s + (1.49 − 2.05i)15-s + (−0.5 − 0.866i)16-s + (−0.0601 − 0.224i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.633 + 0.169i)3-s + (0.249 − 0.433i)4-s + (0.359 − 0.933i)5-s + (0.448 − 0.120i)6-s + (−0.961 + 1.66i)7-s − 0.353i·8-s + (−0.493 − 0.284i)9-s + (−0.109 − 0.698i)10-s + (−0.685 − 0.183i)11-s + (0.231 − 0.231i)12-s + (0.983 + 0.181i)13-s + 1.35i·14-s + (0.386 − 0.530i)15-s + (−0.125 − 0.216i)16-s + (−0.0145 − 0.0544i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.886 + 0.462i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ 0.886 + 0.462i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58060 - 0.387809i\)
\(L(\frac12)\) \(\approx\) \(1.58060 - 0.387809i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.803 + 2.08i)T \)
13 \( 1 + (-3.54 - 0.653i)T \)
good3 \( 1 + (-1.09 - 0.294i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (2.54 - 4.40i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.27 + 0.609i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.0601 + 0.224i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.53 - 5.71i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.674 + 2.51i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.64 - 1.52i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.45 + 4.45i)T + 31iT^{2} \)
37 \( 1 + (-2.49 - 4.31i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.412 - 1.53i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.66 + 0.447i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 + (-5.79 + 5.79i)T - 53iT^{2} \)
59 \( 1 + (2.68 - 0.720i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.23 + 7.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.150 + 0.0866i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.17 + 1.11i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 7.51iT - 73T^{2} \)
79 \( 1 - 5.17iT - 79T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 + (-1.49 + 5.59i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (7.02 + 4.05i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.08133194455526302388324256140, −12.46053952825172938381273692989, −11.50924147289549130490781428863, −9.901454574393125541814435260941, −9.055975842286224243405728905564, −8.322683568142522724767850059840, −6.04979889500919811423842239526, −5.51994418430440175992920374452, −3.64026304453291543733098431316, −2.37304748355670007401130697669, 2.83214390743676487075129984002, 3.82513555513238304471628745603, 5.72477393590659082926599190217, 7.02614059262635501161455266155, 7.57129780219018880010947535711, 9.170898195401335794731381575764, 10.52700787279612328698549312873, 11.10305822302115922443893402330, 12.99740057871187687299675670481, 13.59673160144605244833110390181

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