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Properties

Label 1-2523-2523.353-r1-0-0
Degree $1$
Conductor $2523$
Sign $0.426 - 0.904i$
Analytic cond. $271.134$
Root an. cond. $271.134$
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  + (−0.611 + 0.791i)2-s + (−0.252 − 0.967i)4-s + (−0.757 − 0.653i)5-s + (−0.635 + 0.772i)7-s + (0.920 + 0.391i)8-s + (0.979 − 0.199i)10-s + (−0.131 − 0.991i)11-s + (0.297 − 0.954i)13-s + (−0.222 − 0.974i)14-s + (−0.872 + 0.488i)16-s + (−0.561 + 0.827i)17-s + (0.454 + 0.890i)19-s + (−0.440 + 0.897i)20-s + (0.864 + 0.502i)22-s + (−0.522 − 0.852i)23-s + ⋯
L(s)  = 1  + (−0.611 + 0.791i)2-s + (−0.252 − 0.967i)4-s + (−0.757 − 0.653i)5-s + (−0.635 + 0.772i)7-s + (0.920 + 0.391i)8-s + (0.979 − 0.199i)10-s + (−0.131 − 0.991i)11-s + (0.297 − 0.954i)13-s + (−0.222 − 0.974i)14-s + (−0.872 + 0.488i)16-s + (−0.561 + 0.827i)17-s + (0.454 + 0.890i)19-s + (−0.440 + 0.897i)20-s + (0.864 + 0.502i)22-s + (−0.522 − 0.852i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $0.426 - 0.904i$
Analytic conductor: \(271.134\)
Root analytic conductor: \(271.134\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2523} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2523,\ (1:\ ),\ 0.426 - 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5923761970 - 0.3756407635i\)
\(L(\frac12)\) \(\approx\) \(0.5923761970 - 0.3756407635i\)
\(L(1)\) \(\approx\) \(0.5931028092 + 0.08048937000i\)
\(L(1)\) \(\approx\) \(0.5931028092 + 0.08048937000i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.611 + 0.791i)T \)
5 \( 1 + (-0.757 - 0.653i)T \)
7 \( 1 + (-0.635 + 0.772i)T \)
11 \( 1 + (-0.131 - 0.991i)T \)
13 \( 1 + (0.297 - 0.954i)T \)
17 \( 1 + (-0.561 + 0.827i)T \)
19 \( 1 + (0.454 + 0.890i)T \)
23 \( 1 + (-0.522 - 0.852i)T \)
31 \( 1 + (0.426 - 0.904i)T \)
37 \( 1 + (0.341 - 0.939i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.598 + 0.800i)T \)
47 \( 1 + (0.495 - 0.868i)T \)
53 \( 1 + (-0.931 + 0.362i)T \)
59 \( 1 + (-0.647 + 0.762i)T \)
61 \( 1 + (-0.864 - 0.502i)T \)
67 \( 1 + (0.670 + 0.741i)T \)
71 \( 1 + (0.937 + 0.348i)T \)
73 \( 1 + (0.311 - 0.950i)T \)
79 \( 1 + (0.535 - 0.844i)T \)
83 \( 1 + (0.0695 - 0.997i)T \)
89 \( 1 + (-0.00773 + 0.999i)T \)
97 \( 1 + (-0.573 + 0.818i)T \)
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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

−19.53235115674551838247538216202, −18.73971065652662490076700686218, −18.08530384123628894895579969639, −17.46878682795808752008779155582, −16.6338522366444387009431642962, −15.80126006644828356377965681979, −15.46998314465104240367894574969, −13.96630892563290734487941841226, −13.794525820295226109585169775513, −12.6886395996497458573688004862, −12.06966919801604111252384792488, −11.22467479035945577522318804861, −10.91132053374633645775341009751, −9.73896659663825475740335110047, −9.59101740547440724981273529286, −8.49237538831294102761207543705, −7.540811277429812020494872569506, −7.07738455433016507631688852391, −6.51016861789335910861197516353, −4.75599409632751401910185314426, −4.26130207624840812258004545675, −3.38117953926680583991975871165, −2.73772804181304217397604800391, −1.73565042814947044868128045238, −0.64156610317660390235298102272, 0.26341025264053622666239188123, 0.958082146117195420605753756760, 2.203155859081835682673838677511, 3.36235456351476782348604640631, 4.19524498136642694693182282168, 5.22908772552257583467554986413, 5.97435669310992479901130616058, 6.34338214831168630024121379999, 7.748665357653007084945873157640, 8.06090739820832819974210444258, 8.785170135482503043755726868930, 9.38262610440668072155605254008, 10.364200276567822577411122198306, 11.009988808969536065321092649855, 11.944139747846666071808336285055, 12.776158627750298369110222448720, 13.33919143570087887796696558146, 14.36114738850872935465465847379, 15.17615423129212395517193689370, 15.683826126182688043791343622787, 16.29931285909360776912767465466, 16.74331982815038729945014993979, 17.711287544300735711765599789267, 18.52027905069440531850117206378, 18.93895274987043074158928577664

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