Properties

Label 1-260-260.127-r0-0-0
Degree $1$
Conductor $260$
Sign $0.874 - 0.485i$
Analytic cond. $1.20743$
Root an. cond. $1.20743$
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  + (−0.866 − 0.5i)3-s + (0.866 − 0.5i)7-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s − 21-s + (−0.866 − 0.5i)23-s i·27-s + (0.5 − 0.866i)29-s + 31-s + (0.866 − 0.5i)33-s + (0.866 + 0.5i)37-s + (0.5 − 0.866i)41-s + (0.866 − 0.5i)43-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (0.866 − 0.5i)7-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s − 21-s + (−0.866 − 0.5i)23-s i·27-s + (0.5 − 0.866i)29-s + 31-s + (0.866 − 0.5i)33-s + (0.866 + 0.5i)37-s + (0.5 − 0.866i)41-s + (0.866 − 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.874 - 0.485i$
Analytic conductor: \(1.20743\)
Root analytic conductor: \(1.20743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 260,\ (0:\ ),\ 0.874 - 0.485i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9784674069 - 0.2532061927i\)
\(L(\frac12)\) \(\approx\) \(0.9784674069 - 0.2532061927i\)
\(L(1)\) \(\approx\) \(0.9077892554 - 0.1442782781i\)
\(L(1)\) \(\approx\) \(0.9077892554 - 0.1442782781i\)

Euler product

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

−26.21725085169859389049401282966, −24.88611246082565552647578245458, −23.917763135261806986054582338822, −23.40875710793788191975355533004, −22.06673275423665000814882093837, −21.53063864608709471749294988526, −20.807401710395696679911142044989, −19.474952982180512693871069381383, −18.256456726305331541223844120463, −17.75044922405493533555558864968, −16.60567330403616826958622392147, −15.826469531629766994691061758415, −14.908118687868812886603667484945, −13.82736206807413394617818030151, −12.50429477467068802034265277120, −11.59570493180116825473564882778, −10.884242360779648261713161157516, −9.84790313365915777564299920070, −8.66062801443701439788287227515, −7.584223053601555623999583691032, −6.08107366822137129154632221421, −5.376057262575232269654230288, −4.33844012288193807387749355922, −2.92847322527736990406610441591, −1.147608783590174361403319343733, 1.05148758242879048703436585891, 2.31030587592929068446473001728, 4.23372926085173032092404743981, 5.13622902033099253340118794419, 6.21316269307819146679988215911, 7.518690324482238121394641976671, 7.979590946791762763574337687877, 9.86537589203947134184724768944, 10.57792369036773387109031099303, 11.76619457758433848008631097603, 12.34130939828071595035973145623, 13.589249348355201714126000779290, 14.42314369861750074958692200923, 15.7389452318837831895162978378, 16.67935390136101261654790364586, 17.61327800099531850969888973293, 18.20118919721379238932032034851, 19.18149770729696465126208876305, 20.5082542166236833729444038816, 21.11577481517626341731309974676, 22.487845975055176344686989043921, 23.07785947295063932315205234472, 23.93656945146979434111970909802, 24.70536742159565297632520353199, 25.69913456806159381450227411418

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