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Properties

Label 2-260-260.103-c1-0-7
Degree $2$
Conductor $260$
Sign $0.677 - 0.735i$
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  + (−0.693 − 1.23i)2-s + (0.828 − 0.828i)3-s + (−1.03 + 1.70i)4-s + (−1.31 + 1.80i)5-s + (−1.59 − 0.446i)6-s + (−2.94 + 2.94i)7-s + (2.82 + 0.0923i)8-s + 1.62i·9-s + (3.14 + 0.373i)10-s + 3.40·11-s + (0.556 + 2.27i)12-s + (−1.99 + 3.00i)13-s + (5.66 + 1.58i)14-s + (0.402 + 2.58i)15-s + (−1.84 − 3.54i)16-s + (−2.96 + 2.96i)17-s + ⋯
L(s)  = 1  + (−0.490 − 0.871i)2-s + (0.478 − 0.478i)3-s + (−0.518 + 0.854i)4-s + (−0.590 + 0.807i)5-s + (−0.651 − 0.182i)6-s + (−1.11 + 1.11i)7-s + (0.999 + 0.0326i)8-s + 0.542i·9-s + (0.992 + 0.118i)10-s + 1.02·11-s + (0.160 + 0.656i)12-s + (−0.552 + 0.833i)13-s + (1.51 + 0.423i)14-s + (0.103 + 0.668i)15-s + (−0.461 − 0.886i)16-s + (−0.719 + 0.719i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.677 - 0.735i$
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: \((2,\ 260,\ (\ :1/2),\ 0.677 - 0.735i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.670436 + 0.293968i\)
\(L(\frac12)\) \(\approx\) \(0.670436 + 0.293968i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.693 + 1.23i)T \)
5 \( 1 + (1.31 - 1.80i)T \)
13 \( 1 + (1.99 - 3.00i)T \)
good3 \( 1 + (-0.828 + 0.828i)T - 3iT^{2} \)
7 \( 1 + (2.94 - 2.94i)T - 7iT^{2} \)
11 \( 1 - 3.40T + 11T^{2} \)
17 \( 1 + (2.96 - 2.96i)T - 17iT^{2} \)
19 \( 1 + 7.33iT - 19T^{2} \)
23 \( 1 + (1.65 - 1.65i)T - 23iT^{2} \)
29 \( 1 - 1.98iT - 29T^{2} \)
31 \( 1 - 4.43T + 31T^{2} \)
37 \( 1 + (-1.88 - 1.88i)T + 37iT^{2} \)
41 \( 1 - 7.55iT - 41T^{2} \)
43 \( 1 + (0.548 - 0.548i)T - 43iT^{2} \)
47 \( 1 + (-5.58 + 5.58i)T - 47iT^{2} \)
53 \( 1 + (0.437 + 0.437i)T + 53iT^{2} \)
59 \( 1 - 6.14iT - 59T^{2} \)
61 \( 1 - 8.12T + 61T^{2} \)
67 \( 1 + (3.78 - 3.78i)T - 67iT^{2} \)
71 \( 1 + 8.90T + 71T^{2} \)
73 \( 1 + (-4.20 + 4.20i)T - 73iT^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 + (-0.688 - 0.688i)T + 83iT^{2} \)
89 \( 1 - 17.5T + 89T^{2} \)
97 \( 1 + (3.63 + 3.63i)T + 97iT^{2} \)
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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

−11.88773632936662523055407799601, −11.38703293618718336941799897454, −10.18004795962267465915619863324, −9.154463001205010401672621280682, −8.562883075902598745285092257459, −7.25058118359286310060824002256, −6.50047129959719344575693084647, −4.42573095529326715828413947163, −3.05797066269598576373835397220, −2.22548518066667618321622091270, 0.64105081581283479847790068929, 3.68023329314234225015989432199, 4.40972298848047442479869042979, 6.00780511432399980066268193337, 7.03111980130632840396352519926, 7.994840868020821097736607227673, 9.017948926827994067302007636972, 9.712488888317491220456633076522, 10.40482416259023438467113232270, 11.97944264511894022762109100822

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