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Properties

Label 2-260-20.3-c1-0-18
Degree $2$
Conductor $260$
Sign $0.999 + 0.00293i$
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  + (−0.774 + 1.18i)2-s + (−2.05 + 2.05i)3-s + (−0.800 − 1.83i)4-s + (1.34 − 1.78i)5-s + (−0.841 − 4.03i)6-s + (−0.845 − 0.845i)7-s + (2.78 + 0.472i)8-s − 5.48i·9-s + (1.06 + 2.97i)10-s − 4.19i·11-s + (5.42 + 2.12i)12-s + (−0.707 − 0.707i)13-s + (1.65 − 0.345i)14-s + (0.901 + 6.44i)15-s + (−2.71 + 2.93i)16-s + (−0.206 + 0.206i)17-s + ⋯
L(s)  = 1  + (−0.547 + 0.836i)2-s + (−1.18 + 1.18i)3-s + (−0.400 − 0.916i)4-s + (0.602 − 0.798i)5-s + (−0.343 − 1.64i)6-s + (−0.319 − 0.319i)7-s + (0.985 + 0.167i)8-s − 1.82i·9-s + (0.337 + 0.941i)10-s − 1.26i·11-s + (1.56 + 0.613i)12-s + (−0.196 − 0.196i)13-s + (0.442 − 0.0923i)14-s + (0.232 + 1.66i)15-s + (−0.679 + 0.733i)16-s + (−0.0499 + 0.0499i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.999 + 0.00293i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.999 + 0.00293i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.566255 - 0.000831950i\)
\(L(\frac12)\) \(\approx\) \(0.566255 - 0.000831950i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.774 - 1.18i)T \)
5 \( 1 + (-1.34 + 1.78i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (2.05 - 2.05i)T - 3iT^{2} \)
7 \( 1 + (0.845 + 0.845i)T + 7iT^{2} \)
11 \( 1 + 4.19iT - 11T^{2} \)
17 \( 1 + (0.206 - 0.206i)T - 17iT^{2} \)
19 \( 1 - 4.84T + 19T^{2} \)
23 \( 1 + (0.389 - 0.389i)T - 23iT^{2} \)
29 \( 1 - 6.55iT - 29T^{2} \)
31 \( 1 + 5.06iT - 31T^{2} \)
37 \( 1 + (-8.49 + 8.49i)T - 37iT^{2} \)
41 \( 1 + 5.61T + 41T^{2} \)
43 \( 1 + (-3.25 + 3.25i)T - 43iT^{2} \)
47 \( 1 + (8.91 + 8.91i)T + 47iT^{2} \)
53 \( 1 + (-0.0714 - 0.0714i)T + 53iT^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 + 3.57T + 61T^{2} \)
67 \( 1 + (-6.95 - 6.95i)T + 67iT^{2} \)
71 \( 1 + 4.61iT - 71T^{2} \)
73 \( 1 + (-2.57 - 2.57i)T + 73iT^{2} \)
79 \( 1 + 2.19T + 79T^{2} \)
83 \( 1 + (11.1 - 11.1i)T - 83iT^{2} \)
89 \( 1 - 15.0iT - 89T^{2} \)
97 \( 1 + (5.51 - 5.51i)T - 97iT^{2} \)
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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

−11.63181415684370302442764899557, −10.78978816278364779260302872092, −9.907846947599346846649114492903, −9.342384968800036535590215790766, −8.297538676335046005580516293543, −6.74714682613353628035621853777, −5.60025373408568824905914174672, −5.31655702930851875358502476988, −3.95156969321235186216982376215, −0.65499605642961199413617771103, 1.54976782513604923524428863150, 2.75043466953737821732401703380, 4.83517946917115581069222147658, 6.19854400674361578317979571883, 7.06044722845690837387085446695, 7.84308048321951607412019157612, 9.559021621833022404993673420225, 10.12640970126404380270498489960, 11.31627720742801399318103427363, 11.80516327174496059483386731925

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