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Properties

Label 2-130-65.59-c2-0-13
Degree $2$
Conductor $130$
Sign $0.0255 - 0.999i$
Analytic cond. $3.54224$
Root an. cond. $1.88208$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.366 − 1.36i)2-s + (−4.71 − 2.72i)3-s + (−1.73 + i)4-s + (−3.09 − 3.92i)5-s + (−1.99 + 7.43i)6-s + (1.57 − 5.86i)7-s + (2 + 1.99i)8-s + (10.2 + 17.8i)9-s + (−4.22 + 5.66i)10-s + (1.84 + 6.89i)11-s + 10.8·12-s + (−10.8 + 7.23i)13-s − 8.59·14-s + (3.91 + 26.9i)15-s + (1.99 − 3.46i)16-s + (1.53 + 2.65i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−1.57 − 0.906i)3-s + (−0.433 + 0.250i)4-s + (−0.619 − 0.785i)5-s + (−0.331 + 1.23i)6-s + (0.224 − 0.838i)7-s + (0.250 + 0.249i)8-s + (1.14 + 1.98i)9-s + (−0.422 + 0.566i)10-s + (0.168 + 0.627i)11-s + 0.906·12-s + (−0.831 + 0.556i)13-s − 0.613·14-s + (0.260 + 1.79i)15-s + (0.124 − 0.216i)16-s + (0.0900 + 0.155i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0255 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0255 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.0255 - 0.999i$
Analytic conductor: \(3.54224\)
Root analytic conductor: \(1.88208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1),\ 0.0255 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0949693 + 0.0925750i\)
\(L(\frac12)\) \(\approx\) \(0.0949693 + 0.0925750i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
5 \( 1 + (3.09 + 3.92i)T \)
13 \( 1 + (10.8 - 7.23i)T \)
good3 \( 1 + (4.71 + 2.72i)T + (4.5 + 7.79i)T^{2} \)
7 \( 1 + (-1.57 + 5.86i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (-1.84 - 6.89i)T + (-104. + 60.5i)T^{2} \)
17 \( 1 + (-1.53 - 2.65i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-8.55 + 31.9i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (11.7 - 20.3i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (2.88 - 4.99i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (26.4 + 26.4i)T + 961iT^{2} \)
37 \( 1 + (37.4 - 10.0i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (58.9 - 15.7i)T + (1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (7.65 + 13.2i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-53.1 - 53.1i)T + 2.20e3iT^{2} \)
53 \( 1 + 0.843iT - 2.80e3T^{2} \)
59 \( 1 + (31.3 + 8.38i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-24.7 - 42.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (29.9 + 111. i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (4.97 - 18.5i)T + (-4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (33.1 + 33.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 66.7T + 6.24e3T^{2} \)
83 \( 1 + (5.35 - 5.35i)T - 6.88e3iT^{2} \)
89 \( 1 + (-0.391 - 1.45i)T + (-6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-93.9 - 25.1i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−12.02941277528796998000869422836, −11.57040722229664701004497111012, −10.61768792796746276665913714730, −9.342051089411981126292232313853, −7.63448532704399975159911805425, −7.02549319805495932996659343075, −5.23374042244893008644859505088, −4.36409793803537959320626839878, −1.53036191550263430267663725366, −0.12162535648457572313679358034, 3.73813396325009245787997706621, 5.21872872768652347995898990604, 5.94305871377591585943872849739, 7.11970505740893953518672224234, 8.532900083947881589847808866439, 10.03052201340285207127633208166, 10.59907617638129642718289486606, 11.83777202481838922580365466257, 12.27464502960497379719001348431, 14.33125620928116547306498589576

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