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Properties

Label 1-265-265.219-r0-0-0
Degree $1$
Conductor $265$
Sign $0.963 - 0.267i$
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  + (0.120 − 0.992i)2-s + (0.885 + 0.464i)3-s + (−0.970 − 0.239i)4-s + (0.568 − 0.822i)6-s + (−0.120 + 0.992i)7-s + (−0.354 + 0.935i)8-s + (0.568 + 0.822i)9-s + (−0.748 − 0.663i)11-s + (−0.748 − 0.663i)12-s + (0.970 − 0.239i)13-s + (0.970 + 0.239i)14-s + (0.885 + 0.464i)16-s + (0.354 + 0.935i)17-s + (0.885 − 0.464i)18-s + (0.970 − 0.239i)19-s + ⋯
L(s)  = 1  + (0.120 − 0.992i)2-s + (0.885 + 0.464i)3-s + (−0.970 − 0.239i)4-s + (0.568 − 0.822i)6-s + (−0.120 + 0.992i)7-s + (−0.354 + 0.935i)8-s + (0.568 + 0.822i)9-s + (−0.748 − 0.663i)11-s + (−0.748 − 0.663i)12-s + (0.970 − 0.239i)13-s + (0.970 + 0.239i)14-s + (0.885 + 0.464i)16-s + (0.354 + 0.935i)17-s + (0.885 − 0.464i)18-s + (0.970 − 0.239i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(265\)    =    \(5 \cdot 53\)
Sign: $0.963 - 0.267i$
Analytic conductor: \(1.23065\)
Root analytic conductor: \(1.23065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{265} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 265,\ (0:\ ),\ 0.963 - 0.267i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.560432135 - 0.2122705892i\)
\(L(\frac12)\) \(\approx\) \(1.560432135 - 0.2122705892i\)
\(L(1)\) \(\approx\) \(1.320547693 - 0.2656319856i\)
\(L(1)\) \(\approx\) \(1.320547693 - 0.2656319856i\)

Euler product

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

−25.80768060733420218780407582771, −25.02904768312167639365528552115, −24.14254484189988476987444489518, −23.23610923490228156617566796274, −22.746547142504411528077714854534, −20.994991380029865093745394841767, −20.57279838246603046193518833996, −19.23131193325046812292029898254, −18.37161000701408573725818217243, −17.62055541368300851214437628618, −16.396072773115810904083996849266, −15.59618391209875669138642817272, −14.618745641044707566342857273505, −13.54819668748390923830371709908, −13.370427718592826303187049357286, −12.01601017228484190751634787960, −10.2670464680217808108778928040, −9.36057191314950139704233376539, −8.28135539717008889647377823645, −7.351066660706234762227463638846, −6.82300788301336319868315714004, −5.30235757093850921289011996143, −4.05840465491608677943663077836, −3.06077320227832877544160475118, −1.13234833186865744305370268260, 1.57659051862523565943775889279, 2.90806690492416640814731249027, 3.452443374950621915341588696517, 4.91191406285637406463634375080, 5.85811796343231345501597567763, 7.95787745465662039404221714466, 8.70521107017254174594003729185, 9.52193252747802812806150734493, 10.59822712379850158227391930703, 11.416914908964991377959191530270, 12.81886270748164994553103286262, 13.36022244787385789325805294782, 14.48780439618463829704716269833, 15.35262346865462764518762705083, 16.29456248438944771593359098230, 17.89769811688155097924885398320, 18.81289552345580482031712075294, 19.29320916956282581124482236726, 20.52250327708967512347182672319, 21.11487566844606729995258294895, 21.809637476996392569859455638565, 22.72090215282695925715195722355, 23.92592885054793593268911162122, 24.95498083030992910022998064730, 26.084707134429875271178255261085

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