↑

Properties

Label 1-2624-2624.1243-r0-0-0
Degree $1$
Conductor $2624$
Sign $0.935 - 0.352i$
Analytic cond. $12.1858$
Root an. cond. $12.1858$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.382 − 0.923i)3-s + (−0.760 − 0.649i)5-s + (−0.587 + 0.809i)7-s + (−0.707 − 0.707i)9-s + (−0.649 − 0.760i)11-s + (−0.972 + 0.233i)13-s + (−0.891 + 0.453i)15-s + (−0.453 + 0.891i)17-s + (−0.852 + 0.522i)19-s + (0.522 + 0.852i)21-s + (−0.156 + 0.987i)23-s + (0.156 + 0.987i)25-s + (−0.923 + 0.382i)27-s + (−0.760 − 0.649i)29-s + (−0.309 − 0.951i)31-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)3-s + (−0.760 − 0.649i)5-s + (−0.587 + 0.809i)7-s + (−0.707 − 0.707i)9-s + (−0.649 − 0.760i)11-s + (−0.972 + 0.233i)13-s + (−0.891 + 0.453i)15-s + (−0.453 + 0.891i)17-s + (−0.852 + 0.522i)19-s + (0.522 + 0.852i)21-s + (−0.156 + 0.987i)23-s + (0.156 + 0.987i)25-s + (−0.923 + 0.382i)27-s + (−0.760 − 0.649i)29-s + (−0.309 − 0.951i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $0.935 - 0.352i$
Analytic conductor: \(12.1858\)
Root analytic conductor: \(12.1858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (1243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ 0.935 - 0.352i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6903985531 - 0.1256499054i\)
\(L(\frac12)\) \(\approx\) \(0.6903985531 - 0.1256499054i\)
\(L(1)\) \(\approx\) \(0.7051813849 - 0.2322602369i\)
\(L(1)\) \(\approx\) \(0.7051813849 - 0.2322602369i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 \)
good3 \( 1 + (0.382 - 0.923i)T \)
5 \( 1 + (-0.760 - 0.649i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (-0.649 - 0.760i)T \)
13 \( 1 + (-0.972 + 0.233i)T \)
17 \( 1 + (-0.453 + 0.891i)T \)
19 \( 1 + (-0.852 + 0.522i)T \)
23 \( 1 + (-0.156 + 0.987i)T \)
29 \( 1 + (-0.760 - 0.649i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (0.649 - 0.760i)T \)
43 \( 1 + (0.972 - 0.233i)T \)
47 \( 1 + (0.987 + 0.156i)T \)
53 \( 1 + (-0.0784 - 0.996i)T \)
59 \( 1 + (-0.972 + 0.233i)T \)
61 \( 1 + (0.972 + 0.233i)T \)
67 \( 1 + (-0.649 + 0.760i)T \)
71 \( 1 + (0.951 - 0.309i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (-0.923 + 0.382i)T \)
89 \( 1 + (-0.587 + 0.809i)T \)
97 \( 1 + (0.453 + 0.891i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.67925718924006871530411285755, −18.78847201507490546986547627109, −18.02389086796862988375199427561, −17.04755350495870152017605369246, −16.51766408573503525905562356088, −15.60631992781955127904740388258, −15.33411812881233949559949406566, −14.46385645590383198850249542168, −13.971572974917429741801314605847, −12.90040332743967244791432666804, −12.31807683659264347423635778283, −11.15099025921927343519988233697, −10.69007198930791631525782314233, −10.091577076495081990433625175328, −9.38678239701813838729486203025, −8.52289277539146535165628137091, −7.497647769822755129088490556687, −7.212970056597584914972376643785, −6.22651199322707691372792240385, −4.84243403815625144882804614741, −4.56954579917834013258198556798, −3.66092726933408314016268123278, −2.81808929078467000274544660657, −2.32725103149719542429486009064, −0.36007255527935629726489826724, 0.56095268147066243138404012412, 1.92944743290442177391237163441, 2.47854909264231336953001511283, 3.53371272588335188881173672222, 4.20220721643052298138251788323, 5.57881880667837354635790051292, 5.87824149336293013237692156141, 6.95856708202082371785843313395, 7.79893038765153183366769674750, 8.22971344843133095945574711012, 9.05024456846208197298857846188, 9.59191196197906422041626965842, 10.88427319787901898185299386779, 11.620095064049817453706856411744, 12.30284389642542663033933831976, 12.90146776886550793622642451635, 13.27479449489452662835536323774, 14.3710416880697613933202177173, 15.14749924131876832409578106170, 15.593564545931690257038820879354, 16.5991979945674356752009612508, 17.1164112614269651857255513093, 18.03110747183738603346958288879, 18.98120987429346836695027715699, 19.1777443517606123263364188858

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