↑

Properties

Label 2-130-65.7-c1-0-2
Degree $2$
Conductor $130$
Sign $0.448 - 0.893i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.250 + 0.935i)3-s + (0.499 + 0.866i)4-s + (−2.23 − 0.0997i)5-s + (−0.250 + 0.935i)6-s + (1.88 + 3.26i)7-s + 0.999i·8-s + (1.78 − 1.03i)9-s + (−1.88 − 1.20i)10-s + (−0.866 − 3.23i)11-s + (−0.685 + 0.685i)12-s + (1.65 − 3.20i)13-s + 3.76i·14-s + (−0.466 − 2.11i)15-s + (−0.5 + 0.866i)16-s + (−5.78 − 1.54i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.144 + 0.540i)3-s + (0.249 + 0.433i)4-s + (−0.999 − 0.0446i)5-s + (−0.102 + 0.382i)6-s + (0.712 + 1.23i)7-s + 0.353i·8-s + (0.594 − 0.343i)9-s + (−0.595 − 0.380i)10-s + (−0.261 − 0.974i)11-s + (−0.197 + 0.197i)12-s + (0.458 − 0.888i)13-s + 1.00i·14-s + (−0.120 − 0.546i)15-s + (−0.125 + 0.216i)16-s + (−1.40 − 0.375i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.448 - 0.893i$
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: \((2,\ 130,\ (\ :1/2),\ 0.448 - 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21293 + 0.748659i\)
\(L(\frac12)\) \(\approx\) \(1.21293 + 0.748659i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.0997i)T \)
13 \( 1 + (-1.65 + 3.20i)T \)
good3 \( 1 + (-0.250 - 0.935i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.88 - 3.26i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.866 + 3.23i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (5.78 + 1.54i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.47 + 0.664i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-5.49 + 1.47i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.64 + 0.951i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.67 + 2.67i)T + 31iT^{2} \)
37 \( 1 + (4.85 - 8.40i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.72 - 2.06i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.435 + 1.62i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 12.7T + 47T^{2} \)
53 \( 1 + (0.161 - 0.161i)T - 53iT^{2} \)
59 \( 1 + (0.0190 - 0.0711i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.706 + 1.22i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.12 + 2.95i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.27 - 8.47i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 6.27iT - 73T^{2} \)
79 \( 1 - 1.00iT - 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 + (6.26 - 1.67i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (6.40 - 3.69i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.44340150590411255282864558524, −12.53120518828936778074743967511, −11.47946238291044007189493681731, −10.76655532695477831507566177925, −8.854548492273036544722168685413, −8.365093017739799844001565948941, −6.90019218709431596693552954420, −5.44076702084978251785306358301, −4.37166997365186956049312153747, −3.01410138091956350058261150373, 1.77578400439426050168752820119, 4.01646173483925478288921569161, 4.66710728812932866014660958060, 6.97447679570015031516901155751, 7.33126906872060058420201477071, 8.772850203219899759545604155439, 10.58122955345051194311195161271, 11.04101471710219831286952158704, 12.27478510398798260556023903320, 13.11812133271133819143231514101

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