Properties

Label 2-260-260.207-c1-0-31
Degree $2$
Conductor $260$
Sign $-0.321 + 0.946i$
Analytic cond. $2.07611$
Root an. cond. $1.44087$
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  + (0.417 + 1.35i)2-s + (−0.561 − 0.561i)3-s + (−1.65 + 1.12i)4-s + (−2.09 − 0.789i)5-s + (0.524 − 0.993i)6-s + (−1.48 − 1.48i)7-s + (−2.21 − 1.76i)8-s − 2.36i·9-s + (0.194 − 3.15i)10-s − 5.21·11-s + (1.56 + 0.294i)12-s + (1.57 + 3.24i)13-s + (1.38 − 2.63i)14-s + (0.731 + 1.61i)15-s + (1.45 − 3.72i)16-s + (0.793 + 0.793i)17-s + ⋯
L(s)  = 1  + (0.295 + 0.955i)2-s + (−0.324 − 0.324i)3-s + (−0.825 + 0.563i)4-s + (−0.935 − 0.353i)5-s + (0.214 − 0.405i)6-s + (−0.562 − 0.562i)7-s + (−0.782 − 0.622i)8-s − 0.789i·9-s + (0.0614 − 0.998i)10-s − 1.57·11-s + (0.450 + 0.0849i)12-s + (0.436 + 0.899i)13-s + (0.371 − 0.702i)14-s + (0.188 + 0.417i)15-s + (0.364 − 0.931i)16-s + (0.192 + 0.192i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.321 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.321 + 0.946i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.321 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148520 - 0.207272i\)
\(L(\frac12)\) \(\approx\) \(0.148520 - 0.207272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.417 - 1.35i)T \)
5 \( 1 + (2.09 + 0.789i)T \)
13 \( 1 + (-1.57 - 3.24i)T \)
good3 \( 1 + (0.561 + 0.561i)T + 3iT^{2} \)
7 \( 1 + (1.48 + 1.48i)T + 7iT^{2} \)
11 \( 1 + 5.21T + 11T^{2} \)
17 \( 1 + (-0.793 - 0.793i)T + 17iT^{2} \)
19 \( 1 + 0.101iT - 19T^{2} \)
23 \( 1 + (2.99 + 2.99i)T + 23iT^{2} \)
29 \( 1 + 3.15iT - 29T^{2} \)
31 \( 1 + 5.55T + 31T^{2} \)
37 \( 1 + (3.25 - 3.25i)T - 37iT^{2} \)
41 \( 1 + 6.47iT - 41T^{2} \)
43 \( 1 + (-2.45 - 2.45i)T + 43iT^{2} \)
47 \( 1 + (-3.39 - 3.39i)T + 47iT^{2} \)
53 \( 1 + (6.94 - 6.94i)T - 53iT^{2} \)
59 \( 1 + 10.5iT - 59T^{2} \)
61 \( 1 + 1.49T + 61T^{2} \)
67 \( 1 + (-6.37 - 6.37i)T + 67iT^{2} \)
71 \( 1 - 1.73T + 71T^{2} \)
73 \( 1 + (5.87 + 5.87i)T + 73iT^{2} \)
79 \( 1 + 1.43T + 79T^{2} \)
83 \( 1 + (-10.2 + 10.2i)T - 83iT^{2} \)
89 \( 1 + 6.49T + 89T^{2} \)
97 \( 1 + (-6.63 + 6.63i)T - 97iT^{2} \)
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

−12.06817859064852879755598369949, −10.84761917825722063204489579912, −9.565011230120465413506550506947, −8.498113023008898740149530940403, −7.58613937248543917462328517708, −6.78259347772004829186050603714, −5.73047607197745307467442789413, −4.43322022467793083305857335866, −3.44677194566626132864181575419, −0.18130242012484538929690575557, 2.58003529608583565947012806437, 3.60136020696098185862708784280, 5.02493605907481018102099316903, 5.75782743383660428413295300166, 7.60315374716547794045897313162, 8.451239181165600680003251549714, 9.816425921360421049956121948988, 10.67774327511357206660877377069, 11.11730442326475969916140130332, 12.27434213041074903501709907097

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