Properties

Label 1-260-260.179-r1-0-0
Degree $1$
Conductor $260$
Sign $-0.872 - 0.488i$
Analytic cond. $27.9408$
Root an. cond. $27.9408$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s − 21-s + (−0.5 + 0.866i)23-s + 27-s + (−0.5 + 0.866i)29-s + 31-s + (−0.5 − 0.866i)33-s + (−0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s − 21-s + (−0.5 + 0.866i)23-s + 27-s + (−0.5 + 0.866i)29-s + 31-s + (−0.5 − 0.866i)33-s + (−0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.872 - 0.488i$
Analytic conductor: \(27.9408\)
Root analytic conductor: \(27.9408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 260,\ (1:\ ),\ -0.872 - 0.488i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1523552296 + 0.5835382032i\)
\(L(\frac12)\) \(\approx\) \(-0.1523552296 + 0.5835382032i\)
\(L(1)\) \(\approx\) \(0.6798636959 + 0.3809798232i\)
\(L(1)\) \(\approx\) \(0.6798636959 + 0.3809798232i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.75557346196074694114000303510, −24.335737124544114219894248810899, −23.19858592744874922999903937723, −22.82793295638804362714338361845, −21.39093010316908048757721024011, −20.57083709984608607812608702767, −19.425452577794739641796219015805, −18.60311747040356251377175435852, −17.79632298776074252718782515488, −16.77919726225042222222529922977, −16.17126287554210636308150391101, −14.49634358617532637891611865679, −13.781144096575271275896096957052, −12.90102835129457714593993998332, −11.78437218431059236589076065037, −10.97561192112617959275976588016, −10.02850615530129226377404318078, −8.28888558035901104377953133556, −7.71628307264455985298500489192, −6.537693054544142320904438988848, −5.5588359260136902607218494205, −4.34633039606668098765945203172, −2.771848633028950678100265449203, −1.34312685120691394804881584135, −0.20593116045547772811276918329, 1.83444993313955710189602597387, 3.28770032081180768434747541368, 4.641225142731184130706699795099, 5.36863894835302847853239087018, 6.48760796247073842570276753882, 7.976221643211425112707194261075, 9.039239134686433173707545308537, 10.00583930843996880540097511233, 10.946714312595929038830047046302, 11.92405027641570673500938394905, 12.753023066073002914499078382811, 14.2815195490394942637969163722, 15.31153744125686468248638316723, 15.63673328356745084901547048952, 17.07162185153835704186333552748, 17.66369365692374441842176590604, 18.68265175634180261050224401323, 19.9339466695050162641137960897, 20.95780757593298043284298841890, 21.59242420383886086491001460891, 22.39209679865342044391604005221, 23.44600050534408803396817393610, 24.16492385648550905977124712499, 25.60450398746883568268317392906, 26.04874018034103215118871095450

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