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Properties

Label 1-4640-4640.1739-r1-0-0
Degree $1$
Conductor $4640$
Sign $0.555 + 0.831i$
Analytic cond. $498.637$
Root an. cond. $498.637$
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  + (−0.707 + 0.707i)3-s − i·7-s − i·9-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)13-s − 17-s + (−0.707 + 0.707i)19-s + (−0.707 − 0.707i)21-s − i·23-s + (0.707 + 0.707i)27-s + 31-s − 33-s + (−0.707 − 0.707i)37-s − i·39-s − i·41-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s − i·7-s − i·9-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)13-s − 17-s + (−0.707 + 0.707i)19-s + (−0.707 − 0.707i)21-s − i·23-s + (0.707 + 0.707i)27-s + 31-s − 33-s + (−0.707 − 0.707i)37-s − i·39-s − i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4640\)    =    \(2^{5} \cdot 5 \cdot 29\)
Sign: $0.555 + 0.831i$
Analytic conductor: \(498.637\)
Root analytic conductor: \(498.637\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4640} (1739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4640,\ (1:\ ),\ 0.555 + 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.280497445 + 0.6844401443i\)
\(L(\frac12)\) \(\approx\) \(1.280497445 + 0.6844401443i\)
\(L(1)\) \(\approx\) \(0.8062877810 + 0.2591541642i\)
\(L(1)\) \(\approx\) \(0.8062877810 + 0.2591541642i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
29 \( 1 \)
good3 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 - iT \)
11 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 - iT \)
31 \( 1 + T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (0.707 + 0.707i)T \)
61 \( 1 + (-0.707 + 0.707i)T \)
67 \( 1 + (0.707 - 0.707i)T \)
71 \( 1 - iT \)
73 \( 1 - iT \)
79 \( 1 - T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 + T \)
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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

−17.634564843309356857454172034186, −17.3133733818935379278537957710, −16.84066934373771946765894830229, −15.96225239638214782635028851793, −15.49553158876463010903042731700, −14.19948166026425901233931705622, −13.79248661189413411501312966596, −13.30727943647590324975690631339, −12.60263904333636706087641978035, −11.503567706918082369226773210348, −11.36743401122919658118247180328, −10.67744147506534005330420735641, −9.855122777027216457844403004220, −8.8412523943939525455266092688, −8.3411461352139833165656551894, −7.36142468557848968386012026969, −6.68261521399766400296106389470, −6.40725131165489816532639514854, −5.47092756146369382766260294263, −4.502556874631082260892603464065, −4.01508695694663865688344512735, −2.996925397143934536428575255968, −1.88099684680795814402409113945, −1.23217661636339821009867677366, −0.46624120685387422835479325807, 0.453439251218381560154037863164, 1.57232649377197388119974204262, 2.439724463148359458540377946769, 3.4035178417251616441561020006, 4.20706727201428322082260103110, 4.79464834635425705642958038891, 5.61503335131432701986241687554, 6.34199101215330268355836071102, 6.67795293624020207901544314834, 7.96634577288854974519445834409, 8.79619488106159937809032524191, 9.12375119341829469708015443556, 10.16634182270613178286603363924, 10.53691528646349295312745476419, 11.41258994311088951012217605837, 12.012481267542578765340667137278, 12.548826892720402909793764635641, 13.24667689484986398966209005505, 14.401800739739452370007831057561, 14.91850702723914604804257443525, 15.55593900130922305634796460866, 15.95187778712196849429324931592, 16.90236405321338474296496371696, 17.364167623200658537473987844444, 18.15249341075671933759530724979

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