Properties

Label 2-130-65.58-c1-0-0
Degree $2$
Conductor $130$
Sign $-0.0912 - 0.995i$
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 + (−1.75 + 0.471i)3-s + (0.499 + 0.866i)4-s + (−1.28 + 1.82i)5-s + (−1.75 − 0.471i)6-s + (1.48 + 2.57i)7-s + 0.999i·8-s + (0.270 − 0.156i)9-s + (−2.02 + 0.940i)10-s + (4.61 − 1.23i)11-s + (−1.28 − 1.28i)12-s + (−2.51 − 2.58i)13-s + 2.97i·14-s + (1.40 − 3.82i)15-s + (−0.5 + 0.866i)16-s + (0.185 − 0.691i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−1.01 + 0.271i)3-s + (0.249 + 0.433i)4-s + (−0.575 + 0.817i)5-s + (−0.717 − 0.192i)6-s + (0.562 + 0.973i)7-s + 0.353i·8-s + (0.0903 − 0.0521i)9-s + (−0.641 + 0.297i)10-s + (1.39 − 0.372i)11-s + (−0.371 − 0.371i)12-s + (−0.698 − 0.715i)13-s + 0.795i·14-s + (0.361 − 0.986i)15-s + (−0.125 + 0.216i)16-s + (0.0449 − 0.167i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0912 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0912 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $-0.0912 - 0.995i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ -0.0912 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.710253 + 0.778304i\)
\(L(\frac12)\) \(\approx\) \(0.710253 + 0.778304i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.28 - 1.82i)T \)
13 \( 1 + (2.51 + 2.58i)T \)
good3 \( 1 + (1.75 - 0.471i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-1.48 - 2.57i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.61 + 1.23i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.185 + 0.691i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.43 + 5.36i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.28 - 8.53i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.42 - 0.824i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.93 + 2.93i)T - 31iT^{2} \)
37 \( 1 + (0.553 - 0.958i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.45 - 5.44i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-6.00 - 1.60i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 7.27T + 47T^{2} \)
53 \( 1 + (3.61 + 3.61i)T + 53iT^{2} \)
59 \( 1 + (1.72 + 0.462i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.47 + 6.01i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.296 + 0.171i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.04 + 1.08i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 8.44iT - 73T^{2} \)
79 \( 1 - 0.977iT - 79T^{2} \)
83 \( 1 + 1.99T + 83T^{2} \)
89 \( 1 + (-3.73 - 13.9i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (5.43 - 3.13i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.73323085779095927711168953351, −12.20633740255780640303367561873, −11.52431487109808704777536654671, −11.10860476934676347317463848526, −9.453532645099236030441682595227, −8.008194758989017754966851321488, −6.77421847462186739398455203018, −5.73111869980080475471213242896, −4.71910771247798785302544792578, −3.04539779753628967669268379559, 1.20274576890857845419593956663, 4.06416941277506729618634660186, 4.82808486642148436201888897366, 6.29644765230298903679231030357, 7.34059402128494058393285808057, 8.870963825958791461318496830849, 10.30142676259407519623307585202, 11.37060790222105991050819113206, 12.10491804308987453201030151667, 12.57163315773798743813252708950

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