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Properties

Label 2-260-260.19-c1-0-21
Degree $2$
Conductor $260$
Sign $-0.777 + 0.628i$
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  + (−0.796 − 1.16i)2-s + (−0.640 + 1.10i)3-s + (−0.731 + 1.86i)4-s + (−2.15 + 0.592i)5-s + (1.80 − 0.134i)6-s + (0.819 − 3.05i)7-s + (2.75 − 0.627i)8-s + (0.679 + 1.17i)9-s + (2.40 + 2.04i)10-s + (−1.35 − 5.06i)11-s + (−1.59 − 2.00i)12-s + (−3.44 + 1.06i)13-s + (−4.22 + 1.47i)14-s + (0.724 − 2.77i)15-s + (−2.92 − 2.72i)16-s + (−2.66 − 4.61i)17-s + ⋯
L(s)  = 1  + (−0.563 − 0.826i)2-s + (−0.369 + 0.640i)3-s + (−0.365 + 0.930i)4-s + (−0.964 + 0.264i)5-s + (0.737 − 0.0550i)6-s + (0.309 − 1.15i)7-s + (0.975 − 0.221i)8-s + (0.226 + 0.392i)9-s + (0.761 + 0.647i)10-s + (−0.409 − 1.52i)11-s + (−0.460 − 0.578i)12-s + (−0.955 + 0.296i)13-s + (−1.12 + 0.394i)14-s + (0.186 − 0.715i)15-s + (−0.732 − 0.680i)16-s + (−0.645 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 + 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.777 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.777 + 0.628i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.777 + 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.130475 - 0.369067i\)
\(L(\frac12)\) \(\approx\) \(0.130475 - 0.369067i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.796 + 1.16i)T \)
5 \( 1 + (2.15 - 0.592i)T \)
13 \( 1 + (3.44 - 1.06i)T \)
good3 \( 1 + (0.640 - 1.10i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.819 + 3.05i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.35 + 5.06i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.66 + 4.61i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.958 + 3.57i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.21 + 1.27i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.573 + 0.993i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.95 - 1.95i)T + 31iT^{2} \)
37 \( 1 + (-0.831 - 3.10i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.58 + 9.66i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.76 - 1.01i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.80 - 3.80i)T + 47iT^{2} \)
53 \( 1 - 1.42iT - 53T^{2} \)
59 \( 1 + (13.1 + 3.51i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-7.45 - 12.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.53 + 9.46i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.23 - 4.59i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.511 - 0.511i)T - 73iT^{2} \)
79 \( 1 + 6.58iT - 79T^{2} \)
83 \( 1 + (2.29 - 2.29i)T - 83iT^{2} \)
89 \( 1 + (-2.83 + 0.760i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.74 - 10.2i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.23507354747687749657668572230, −10.84690041225069729275886170322, −10.04318685239965146089127289430, −8.834184833575409156724401725781, −7.76067315445025452281862387252, −7.07776286486792429889088936486, −4.88510100566289245198999307704, −4.15019130552606462363474448348, −2.87369709980662876961281522802, −0.38185195995281966913200559872, 1.86967235556236800631410738599, 4.37061912961856369017012230104, 5.43240237641348462787673633554, 6.58146605296035828204908150285, 7.56234536751514563749651949218, 8.178457270680430513348285392593, 9.330557141367603661671593662561, 10.24311283096713612917171939494, 11.62972448166826845423291422502, 12.37079820336529173838173298202

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