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Properties

Label 2-130-65.7-c1-0-3
Degree $2$
Conductor $130$
Sign $0.374 - 0.927i$
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 + (0.706 + 2.63i)3-s + (0.499 + 0.866i)4-s + (0.892 − 2.05i)5-s + (−0.706 + 2.63i)6-s + (−1.79 − 3.11i)7-s + 0.999i·8-s + (−3.84 + 2.22i)9-s + (1.79 − 1.32i)10-s + (0.0725 + 0.270i)11-s + (−1.92 + 1.92i)12-s + (−3.60 − 0.103i)13-s − 3.59i·14-s + (6.03 + 0.904i)15-s + (−0.5 + 0.866i)16-s + (−3.52 − 0.944i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.407 + 1.52i)3-s + (0.249 + 0.433i)4-s + (0.399 − 0.916i)5-s + (−0.288 + 1.07i)6-s + (−0.679 − 1.17i)7-s + 0.353i·8-s + (−1.28 + 0.740i)9-s + (0.568 − 0.420i)10-s + (0.0218 + 0.0816i)11-s + (−0.556 + 0.556i)12-s + (−0.999 − 0.0286i)13-s − 0.961i·14-s + (1.55 + 0.233i)15-s + (−0.125 + 0.216i)16-s + (−0.854 − 0.229i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.374 - 0.927i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ 0.374 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30894 + 0.883345i\)
\(L(\frac12)\) \(\approx\) \(1.30894 + 0.883345i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.892 + 2.05i)T \)
13 \( 1 + (3.60 + 0.103i)T \)
good3 \( 1 + (-0.706 - 2.63i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (1.79 + 3.11i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0725 - 0.270i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.52 + 0.944i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-6.15 - 1.64i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.08 + 0.557i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.424 + 0.244i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.41 - 4.41i)T + 31iT^{2} \)
37 \( 1 + (-0.944 + 1.63i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (8.19 - 2.19i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.18 - 11.9i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 4.74T + 47T^{2} \)
53 \( 1 + (-6.50 + 6.50i)T - 53iT^{2} \)
59 \( 1 + (-3.49 + 13.0i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.444 - 0.769i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.90 - 2.25i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.951 + 3.55i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 1.64iT - 73T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 + (-2.67 + 0.717i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-8.56 + 4.94i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.68903223630643046954923326306, −12.79051951382176048195674734860, −11.43046284828806222390007469915, −9.997790507199405727560804471702, −9.652721349597124218778485693333, −8.329176052837090070360817924624, −6.87656986938134182111059884052, −5.16848609339252966803449276683, −4.45763049297916297223194793842, −3.23840333296428993476056814819, 2.21355713128005205388728381919, 3.00133869840295706742766351989, 5.53145898962557611484248711185, 6.58802907565959240810328981016, 7.33550212129183204903383365834, 8.902508919263414697218640328138, 10.03651187732135336256697932111, 11.62326508130420787725880263269, 12.15720568055620384718678205043, 13.27707224719774048598457277262

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