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Properties

Label 2-130-13.10-c1-0-2
Degree $2$
Conductor $130$
Sign $0.982 + 0.184i$
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 + (0.300 + 0.519i)3-s + (0.499 − 0.866i)4-s + i·5-s + (0.519 + 0.300i)6-s + (1.24 + 0.719i)7-s − 0.999i·8-s + (1.31 − 2.28i)9-s + (0.5 + 0.866i)10-s + (−2.40 + 1.38i)11-s + 0.600·12-s + (−2.76 − 2.31i)13-s + 1.43·14-s + (−0.519 + 0.300i)15-s + (−0.5 − 0.866i)16-s + (−2.25 + 3.90i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.173 + 0.300i)3-s + (0.249 − 0.433i)4-s + 0.447i·5-s + (0.212 + 0.122i)6-s + (0.471 + 0.272i)7-s − 0.353i·8-s + (0.439 − 0.762i)9-s + (0.158 + 0.273i)10-s + (−0.723 + 0.417i)11-s + 0.173·12-s + (−0.767 − 0.641i)13-s + 0.384·14-s + (−0.134 + 0.0774i)15-s + (−0.125 − 0.216i)16-s + (−0.546 + 0.945i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.184i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.982 + 0.184i$
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: \((2,\ 130,\ (\ :1/2),\ 0.982 + 0.184i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51934 - 0.141321i\)
\(L(\frac12)\) \(\approx\) \(1.51934 - 0.141321i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.76 + 2.31i)T \)
good3 \( 1 + (-0.300 - 0.519i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.24 - 0.719i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.40 - 1.38i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.25 - 3.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.75 + 2.16i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.21 - 3.84i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.93 + 6.81i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.16iT - 31T^{2} \)
37 \( 1 + (-0.498 + 0.287i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.65 + 2.11i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.30 + 2.26i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 12.7iT - 47T^{2} \)
53 \( 1 - 9.57T + 53T^{2} \)
59 \( 1 + (3 + 1.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.25 + 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.61 + 2.66i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.19 - 3i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.66iT - 73T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 - 10.0iT - 83T^{2} \)
89 \( 1 + (5.15 - 2.97i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.37 - 0.793i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.08763434931047271879397997537, −12.45080495850787517822922539581, −11.20720870508047221844295920473, −10.35210653628978179622339081717, −9.339578533942430756632526874711, −7.85473741137549258293803665938, −6.57331600948164096410336275367, −5.19169995989333273338293523609, −3.93146907495075853806959967199, −2.39435163768439736016496230323, 2.31917783647021561944753026863, 4.37890393125780927473465158296, 5.27259468043530060122547933144, 6.90993703496799386881139570526, 7.80056086441769491304815450075, 8.867288826965724597157084148625, 10.41803458314527004217145056477, 11.44799355010585667470648334925, 12.65828173229537891766067836136, 13.33458180120631652408297190110

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