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Properties

Label 2-260-20.7-c1-0-15
Degree $2$
Conductor $260$
Sign $0.599 - 0.800i$
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  + (0.867 + 1.11i)2-s + (−0.487 − 0.487i)3-s + (−0.494 + 1.93i)4-s + (2.20 + 0.360i)5-s + (0.121 − 0.966i)6-s + (3.04 − 3.04i)7-s + (−2.59 + 1.12i)8-s − 2.52i·9-s + (1.51 + 2.77i)10-s + 6.32i·11-s + (1.18 − 0.703i)12-s + (−0.707 + 0.707i)13-s + (6.04 + 0.758i)14-s + (−0.899 − 1.25i)15-s + (−3.51 − 1.91i)16-s + (0.541 + 0.541i)17-s + ⋯
L(s)  = 1  + (0.613 + 0.789i)2-s + (−0.281 − 0.281i)3-s + (−0.247 + 0.968i)4-s + (0.986 + 0.161i)5-s + (0.0495 − 0.394i)6-s + (1.15 − 1.15i)7-s + (−0.916 + 0.399i)8-s − 0.841i·9-s + (0.478 + 0.878i)10-s + 1.90i·11-s + (0.342 − 0.203i)12-s + (−0.196 + 0.196i)13-s + (1.61 + 0.202i)14-s + (−0.232 − 0.322i)15-s + (−0.877 − 0.478i)16-s + (0.131 + 0.131i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.599 - 0.800i$
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: \((2,\ 260,\ (\ :1/2),\ 0.599 - 0.800i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64157 + 0.820941i\)
\(L(\frac12)\) \(\approx\) \(1.64157 + 0.820941i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.867 - 1.11i)T \)
5 \( 1 + (-2.20 - 0.360i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (0.487 + 0.487i)T + 3iT^{2} \)
7 \( 1 + (-3.04 + 3.04i)T - 7iT^{2} \)
11 \( 1 - 6.32iT - 11T^{2} \)
17 \( 1 + (-0.541 - 0.541i)T + 17iT^{2} \)
19 \( 1 + 5.33T + 19T^{2} \)
23 \( 1 + (-1.30 - 1.30i)T + 23iT^{2} \)
29 \( 1 + 4.13iT - 29T^{2} \)
31 \( 1 + 8.12iT - 31T^{2} \)
37 \( 1 + (2.03 + 2.03i)T + 37iT^{2} \)
41 \( 1 + 5.86T + 41T^{2} \)
43 \( 1 + (-1.47 - 1.47i)T + 43iT^{2} \)
47 \( 1 + (2.36 - 2.36i)T - 47iT^{2} \)
53 \( 1 + (-1.38 + 1.38i)T - 53iT^{2} \)
59 \( 1 + 6.39T + 59T^{2} \)
61 \( 1 + 6.69T + 61T^{2} \)
67 \( 1 + (-3.14 + 3.14i)T - 67iT^{2} \)
71 \( 1 + 1.54iT - 71T^{2} \)
73 \( 1 + (5.09 - 5.09i)T - 73iT^{2} \)
79 \( 1 - 4.56T + 79T^{2} \)
83 \( 1 + (-6.37 - 6.37i)T + 83iT^{2} \)
89 \( 1 + 3.46iT - 89T^{2} \)
97 \( 1 + (-1.56 - 1.56i)T + 97iT^{2} \)
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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

−12.40425819379192570515749752558, −11.37331480961375992405243773352, −10.16809278720306274631504584094, −9.228999424352784651669218429788, −7.80449245729291319404379489555, −7.04151919881628190392508312338, −6.23224335210441449438192733755, −4.90587318071515032140153849128, −4.10534570194513648692290062839, −1.97208818738737624330600501095, 1.74911033798455214888278561767, 2.94923856771741507792923512992, 4.86730713078497961280348476697, 5.39940191209253468212533121743, 6.25383029722000046785719612940, 8.468336016216939412503243552668, 8.909362635644553043997983940361, 10.44417925060078709465931500855, 10.85845218760353917668424370574, 11.75593391004483090838294454620

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