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Properties

Label 2-260-20.7-c1-0-35
Degree $2$
Conductor $260$
Sign $-0.770 - 0.637i$
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  + (0.904 − 1.08i)2-s + (−1.78 − 1.78i)3-s + (−0.364 − 1.96i)4-s + (−1.34 + 1.78i)5-s + (−3.56 + 0.327i)6-s + (−2.34 + 2.34i)7-s + (−2.46 − 1.38i)8-s + 3.39i·9-s + (0.733 + 3.07i)10-s − 5.90i·11-s + (−2.86 + 4.16i)12-s + (−0.707 + 0.707i)13-s + (0.429 + 4.66i)14-s + (5.59 − 0.803i)15-s + (−3.73 + 1.43i)16-s + (−0.975 − 0.975i)17-s + ⋯
L(s)  = 1  + (0.639 − 0.768i)2-s + (−1.03 − 1.03i)3-s + (−0.182 − 0.983i)4-s + (−0.599 + 0.800i)5-s + (−1.45 + 0.133i)6-s + (−0.886 + 0.886i)7-s + (−0.872 − 0.488i)8-s + 1.13i·9-s + (0.232 + 0.972i)10-s − 1.78i·11-s + (−0.826 + 1.20i)12-s + (−0.196 + 0.196i)13-s + (0.114 + 1.24i)14-s + (1.44 − 0.207i)15-s + (−0.933 + 0.358i)16-s + (−0.236 − 0.236i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.770 - 0.637i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.770 - 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.162183 + 0.450341i\)
\(L(\frac12)\) \(\approx\) \(0.162183 + 0.450341i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.904 + 1.08i)T \)
5 \( 1 + (1.34 - 1.78i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (1.78 + 1.78i)T + 3iT^{2} \)
7 \( 1 + (2.34 - 2.34i)T - 7iT^{2} \)
11 \( 1 + 5.90iT - 11T^{2} \)
17 \( 1 + (0.975 + 0.975i)T + 17iT^{2} \)
19 \( 1 + 2.12T + 19T^{2} \)
23 \( 1 + (-3.10 - 3.10i)T + 23iT^{2} \)
29 \( 1 + 5.56iT - 29T^{2} \)
31 \( 1 + 7.23iT - 31T^{2} \)
37 \( 1 + (2.19 + 2.19i)T + 37iT^{2} \)
41 \( 1 + 0.538T + 41T^{2} \)
43 \( 1 + (5.78 + 5.78i)T + 43iT^{2} \)
47 \( 1 + (3.98 - 3.98i)T - 47iT^{2} \)
53 \( 1 + (-7.09 + 7.09i)T - 53iT^{2} \)
59 \( 1 - 3.05T + 59T^{2} \)
61 \( 1 - 8.82T + 61T^{2} \)
67 \( 1 + (2.58 - 2.58i)T - 67iT^{2} \)
71 \( 1 - 8.53iT - 71T^{2} \)
73 \( 1 + (-2.76 + 2.76i)T - 73iT^{2} \)
79 \( 1 + 7.02T + 79T^{2} \)
83 \( 1 + (-6.93 - 6.93i)T + 83iT^{2} \)
89 \( 1 - 9.42iT - 89T^{2} \)
97 \( 1 + (11.8 + 11.8i)T + 97iT^{2} \)
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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

−11.42683233221069648431565721059, −11.13640197879561708272169862258, −9.817539017324087237972705579298, −8.513875789135884433665014488700, −6.95681993258115120626980838626, −6.16717073106076472783486154103, −5.54492410276289851383778346278, −3.66175644745402116693559563515, −2.50214223503006783684390585541, −0.33059063743270208101684517845, 3.63623084899759788770557214643, 4.60542405179351538907295672878, 5.05507920846573036862181003566, 6.57023668165043660673345970620, 7.29697782824314197050719744904, 8.713947178194216152595169703785, 9.814620084008887268724067466861, 10.62923748511139522450921026967, 11.88019471795216459542470081969, 12.59357660136679172392980443519

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