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Properties

Label 2-130-65.19-c2-0-7
Degree $2$
Conductor $130$
Sign $-0.571 + 0.820i$
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  + (−1.36 + 0.366i)2-s + (−2.81 − 1.62i)3-s + (1.73 − i)4-s + (4.89 + 1.01i)5-s + (4.44 + 1.19i)6-s + (−3.91 − 1.04i)7-s + (−1.99 + 2i)8-s + (0.800 + 1.38i)9-s + (−7.05 + 0.404i)10-s + (−3.41 + 0.915i)11-s − 6.51·12-s + (1.34 − 12.9i)13-s + 5.72·14-s + (−12.1 − 10.8i)15-s + (1.99 − 3.46i)16-s + (−14.4 − 24.9i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.939 − 0.542i)3-s + (0.433 − 0.250i)4-s + (0.979 + 0.203i)5-s + (0.741 + 0.198i)6-s + (−0.559 − 0.149i)7-s + (−0.249 + 0.250i)8-s + (0.0889 + 0.153i)9-s + (−0.705 + 0.0404i)10-s + (−0.310 + 0.0832i)11-s − 0.542·12-s + (0.103 − 0.994i)13-s + 0.409·14-s + (−0.810 − 0.722i)15-s + (0.124 − 0.216i)16-s + (−0.848 − 1.46i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $-0.571 + 0.820i$
Analytic conductor: \(3.54224\)
Root analytic conductor: \(1.88208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1),\ -0.571 + 0.820i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.231473 - 0.443366i\)
\(L(\frac12)\) \(\approx\) \(0.231473 - 0.443366i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
5 \( 1 + (-4.89 - 1.01i)T \)
13 \( 1 + (-1.34 + 12.9i)T \)
good3 \( 1 + (2.81 + 1.62i)T + (4.5 + 7.79i)T^{2} \)
7 \( 1 + (3.91 + 1.04i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (3.41 - 0.915i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (14.4 + 24.9i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (35.3 + 9.48i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (-2.41 + 4.18i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-16.0 + 27.8i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (5.98 - 5.98i)T - 961iT^{2} \)
37 \( 1 + (-10.3 - 38.5i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-4.98 - 18.6i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (9.17 + 15.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-32.9 + 32.9i)T - 2.20e3iT^{2} \)
53 \( 1 + 45.3iT - 2.80e3T^{2} \)
59 \( 1 + (-20.3 + 75.8i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-36.9 - 63.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-22.4 + 6.02i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-94.9 - 25.4i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (71.9 - 71.9i)T - 5.32e3iT^{2} \)
79 \( 1 - 3.01T + 6.24e3T^{2} \)
83 \( 1 + (-7.10 - 7.10i)T + 6.88e3iT^{2} \)
89 \( 1 + (-28.7 + 7.70i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (18.2 - 68.0i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−12.80481274462426810147505257939, −11.49402264987380968397260757387, −10.59141585080561214516293091457, −9.709920988581544898825060268013, −8.513511087658339348882096461467, −6.88664212063295622008206599045, −6.37508070801415795964148186670, −5.18299179236704472558280447201, −2.53068725608298649514771882018, −0.44589712033851348132837041247, 2.07652680824894867330887570268, 4.32293790302541631860450977165, 5.89290449650492735774248449050, 6.55193562137300431771176249975, 8.484522762846339506363553472847, 9.369305421556597519695488466948, 10.56664166670852394097335888659, 10.85220851955228836434695595440, 12.35298391266128515577825759694, 13.10940438821217879107338127899

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