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Properties

Label 2-260-260.103-c1-0-0
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

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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: $\chi_{260} (103, \cdot )$
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 \( 1 + (-6.63 - 6.63i)T + 97iT^{2} \)
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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

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

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