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Properties

Label 1-265-265.253-r0-0-0
Degree $1$
Conductor $265$
Sign $0.590 - 0.807i$
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  + (−0.748 − 0.663i)2-s + (0.970 − 0.239i)3-s + (0.120 + 0.992i)4-s + (−0.885 − 0.464i)6-s + (0.663 − 0.748i)7-s + (0.568 − 0.822i)8-s + (0.885 − 0.464i)9-s + (0.354 + 0.935i)11-s + (0.354 + 0.935i)12-s + (0.992 + 0.120i)13-s + (−0.992 + 0.120i)14-s + (−0.970 + 0.239i)16-s + (−0.822 + 0.568i)17-s + (−0.970 − 0.239i)18-s + (−0.992 − 0.120i)19-s + ⋯
L(s)  = 1  + (−0.748 − 0.663i)2-s + (0.970 − 0.239i)3-s + (0.120 + 0.992i)4-s + (−0.885 − 0.464i)6-s + (0.663 − 0.748i)7-s + (0.568 − 0.822i)8-s + (0.885 − 0.464i)9-s + (0.354 + 0.935i)11-s + (0.354 + 0.935i)12-s + (0.992 + 0.120i)13-s + (−0.992 + 0.120i)14-s + (−0.970 + 0.239i)16-s + (−0.822 + 0.568i)17-s + (−0.970 − 0.239i)18-s + (−0.992 − 0.120i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(265\)    =    \(5 \cdot 53\)
Sign: $0.590 - 0.807i$
Analytic conductor: \(1.23065\)
Root analytic conductor: \(1.23065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{265} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 265,\ (0:\ ),\ 0.590 - 0.807i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.179444001 - 0.5985735474i\)
\(L(\frac12)\) \(\approx\) \(1.179444001 - 0.5985735474i\)
\(L(1)\) \(\approx\) \(1.059231780 - 0.3827693369i\)
\(L(1)\) \(\approx\) \(1.059231780 - 0.3827693369i\)

Euler product

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

−25.84752282600021265980382619139, −25.14028273336250450304750838669, −24.50398095088151208323791498141, −23.63996523870937061573315032804, −22.25307954673199141171082089332, −21.13563799879743931064004512658, −20.42729474389489546977866771154, −19.198520964538921118874549557128, −18.74396653821810256373983815098, −17.74817103315130532111815887866, −16.56235852734435412889234501095, −15.62436821701094530391130390890, −14.98283477126624495028893107395, −14.06961364277270124117146587726, −13.17729483121423193576542385963, −11.3794697546352002005128884499, −10.64108170114785821464783851708, −9.163153888342956951560092590183, −8.74881267291413448442730947127, −7.94513331172620867712120857404, −6.67535940829204074597945681854, −5.5374422776764727930510722234, −4.25770976454136798357164893914, −2.67424776191376220518295170869, −1.45530969789255707123348207507, 1.361102364360063443424483423083, 2.17342496062691162638069909707, 3.68398959666182934302543290840, 4.3862260809499116110941467241, 6.753241785157947666000575030088, 7.49904878244901762264456619244, 8.59333427222082953777054528925, 9.20118490643401695752551934994, 10.49715924403886208829895753896, 11.17871603161967510876414302372, 12.65794601614830533292289398540, 13.22249055942932220808818626960, 14.43058193447947713414004834245, 15.364739315863241901904154824418, 16.67429778836838581459991667424, 17.66023109923997584043519374777, 18.35349654602613274865472249159, 19.47958448844555873616586285838, 20.08144475734489292413928545258, 20.85796279710925639765947090285, 21.53626706599589520857634093795, 22.963033409016535692431211454219, 24.02438345364989851290844513051, 25.11793917160906644419299090537, 25.86841433785689261381636091864

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