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Properties

Label 1-265-265.3-r0-0-0
Degree $1$
Conductor $265$
Sign $0.922 + 0.386i$
Analytic cond. $1.23065$
Root an. cond. $1.23065$
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  + (0.885 + 0.464i)2-s + (0.354 − 0.935i)3-s + (0.568 + 0.822i)4-s + (0.748 − 0.663i)6-s + (−0.464 + 0.885i)7-s + (0.120 + 0.992i)8-s + (−0.748 − 0.663i)9-s + (0.970 − 0.239i)11-s + (0.970 − 0.239i)12-s + (0.822 + 0.568i)13-s + (−0.822 + 0.568i)14-s + (−0.354 + 0.935i)16-s + (0.992 + 0.120i)17-s + (−0.354 − 0.935i)18-s + (−0.822 − 0.568i)19-s + ⋯
L(s)  = 1  + (0.885 + 0.464i)2-s + (0.354 − 0.935i)3-s + (0.568 + 0.822i)4-s + (0.748 − 0.663i)6-s + (−0.464 + 0.885i)7-s + (0.120 + 0.992i)8-s + (−0.748 − 0.663i)9-s + (0.970 − 0.239i)11-s + (0.970 − 0.239i)12-s + (0.822 + 0.568i)13-s + (−0.822 + 0.568i)14-s + (−0.354 + 0.935i)16-s + (0.992 + 0.120i)17-s + (−0.354 − 0.935i)18-s + (−0.822 − 0.568i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(265\)    =    \(5 \cdot 53\)
Sign: $0.922 + 0.386i$
Analytic conductor: \(1.23065\)
Root analytic conductor: \(1.23065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{265} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 265,\ (0:\ ),\ 0.922 + 0.386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.232625880 + 0.4484728002i\)
\(L(\frac12)\) \(\approx\) \(2.232625880 + 0.4484728002i\)
\(L(1)\) \(\approx\) \(1.853266548 + 0.2408510335i\)
\(L(1)\) \(\approx\) \(1.853266548 + 0.2408510335i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
53 \( 1 \)
good2 \( 1 + (0.885 + 0.464i)T \)
3 \( 1 + (0.354 - 0.935i)T \)
7 \( 1 + (-0.464 + 0.885i)T \)
11 \( 1 + (0.970 - 0.239i)T \)
13 \( 1 + (0.822 + 0.568i)T \)
17 \( 1 + (0.992 + 0.120i)T \)
19 \( 1 + (-0.822 - 0.568i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.970 - 0.239i)T \)
31 \( 1 + (0.239 - 0.970i)T \)
37 \( 1 + (-0.935 - 0.354i)T \)
41 \( 1 + (-0.239 - 0.970i)T \)
43 \( 1 + (-0.935 + 0.354i)T \)
47 \( 1 + (0.663 + 0.748i)T \)
59 \( 1 + (-0.748 + 0.663i)T \)
61 \( 1 + (0.992 - 0.120i)T \)
67 \( 1 + (-0.568 - 0.822i)T \)
71 \( 1 + (-0.935 + 0.354i)T \)
73 \( 1 + (-0.120 + 0.992i)T \)
79 \( 1 + (-0.464 - 0.885i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.120 + 0.992i)T \)
97 \( 1 + (-0.663 + 0.748i)T \)
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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

−25.46883723374329396551105725942, −25.11275784011405963342900876576, −23.46043587793708413868470830762, −22.94811350950629811362819497871, −22.146922454389261788770795793565, −21.08889826523515272978161534185, −20.48789985193706830880490575045, −19.70495764943694272463894728315, −18.84771109485909104462757935068, −17.03648208201261920209402084061, −16.36715004784589717003525192509, −15.26392286770144399777921797401, −14.50498890634209218175397214565, −13.68055459608440808530860451228, −12.71278555082717747068887271866, −11.47499198463235884199670009256, −10.505225376092059851517054608771, −9.91328846382331846259950546365, −8.68697579000727095586836544733, −7.13284037778497023121219895546, −5.96948671086359518561577099466, −4.80974350397852361066378456325, −3.71630599553328273540465758118, −3.234853767939250865666371601934, −1.43465827876100189055455506867, 1.72190099730286669398908788269, 2.93958525699130925344544244063, 3.92397699169729036773875967211, 5.6148755644331283541082745889, 6.3454785053664255505729695257, 7.18993368836223418847629597853, 8.48134706732455259722008331535, 9.14845292191803722507434090854, 11.23038550478782489602255587934, 12.005723582372832337903563265331, 12.84797621253114267059530920572, 13.6505300962532851023980070249, 14.6115209126368995709472427682, 15.34283822524384669916690684469, 16.60071593291791744056980233769, 17.36820343786335622339650276529, 18.76286069438090954718909612413, 19.27188925302942836429622240140, 20.572376116149506429172948704092, 21.40580041164348285153683543125, 22.451522118380615770924204210205, 23.231503625776417321201955519054, 24.13640083352647411895241152944, 24.891855565953430483956846362873, 25.61475715592944413761592722815

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