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Properties

Label 1-2624-2624.1219-r0-0-0
Degree $1$
Conductor $2624$
Sign $-0.343 - 0.939i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.382 − 0.923i)3-s + (0.522 − 0.852i)5-s + (0.309 − 0.951i)7-s + (−0.707 + 0.707i)9-s + (0.522 + 0.852i)11-s + (−0.649 − 0.760i)13-s + (−0.987 − 0.156i)15-s + (0.156 + 0.987i)17-s + (0.996 − 0.0784i)19-s + (−0.996 + 0.0784i)21-s + (0.453 − 0.891i)23-s + (−0.453 − 0.891i)25-s + (0.923 + 0.382i)27-s + (0.852 + 0.522i)29-s + (0.809 + 0.587i)31-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)3-s + (0.522 − 0.852i)5-s + (0.309 − 0.951i)7-s + (−0.707 + 0.707i)9-s + (0.522 + 0.852i)11-s + (−0.649 − 0.760i)13-s + (−0.987 − 0.156i)15-s + (0.156 + 0.987i)17-s + (0.996 − 0.0784i)19-s + (−0.996 + 0.0784i)21-s + (0.453 − 0.891i)23-s + (−0.453 − 0.891i)25-s + (0.923 + 0.382i)27-s + (0.852 + 0.522i)29-s + (0.809 + 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $-0.343 - 0.939i$
Analytic conductor: \(12.1858\)
Root analytic conductor: \(12.1858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (1219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ -0.343 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.033462385 - 1.477937747i\)
\(L(\frac12)\) \(\approx\) \(1.033462385 - 1.477937747i\)
\(L(1)\) \(\approx\) \(0.9924029524 - 0.5814526703i\)
\(L(1)\) \(\approx\) \(0.9924029524 - 0.5814526703i\)

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 \( 1 + (0.522 - 0.852i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (0.522 + 0.852i)T \)
13 \( 1 + (-0.649 - 0.760i)T \)
17 \( 1 + (0.156 + 0.987i)T \)
19 \( 1 + (0.996 - 0.0784i)T \)
23 \( 1 + (0.453 - 0.891i)T \)
29 \( 1 + (0.852 + 0.522i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.852 + 0.522i)T \)
43 \( 1 + (0.760 - 0.649i)T \)
47 \( 1 + (-0.453 + 0.891i)T \)
53 \( 1 + (-0.972 + 0.233i)T \)
59 \( 1 + (0.760 - 0.649i)T \)
61 \( 1 + (0.760 + 0.649i)T \)
67 \( 1 + (0.522 - 0.852i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + (0.382 - 0.923i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.156 - 0.987i)T \)
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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

−19.360475335589541281667438719899, −18.90370147751468519663874025604, −17.974656882409186724000562302626, −17.548794705340603409888422439852, −16.6852098730901154017289762333, −15.96495295522160690508379087515, −15.39746594239092650530094794261, −14.385687480146727475225647719778, −14.30598967336044778336935833674, −13.320072699902272224210528354753, −11.97624730188921859360309094578, −11.521149579388227188982115831, −11.18758055656789059218484010191, −9.95825450816121506962724938381, −9.564793315054972423408743505856, −8.98865803024068174338764377271, −7.945846508749253078477518529974, −6.91581407288748564371371695120, −6.15867856543608957568906300312, −5.5093402696503966540631103642, −4.86755301700594651096270341401, −3.821306251129508555271251909208, −2.93523043883024199471527354709, −2.405884323808893886230529619510, −1.01588038557318326290153830117, 0.7733301382582921288690738017, 1.29125384103531500719468188233, 2.172606810258810525943706810136, 3.224277457482222241255380193425, 4.62353994693596002577052348696, 4.85702954511410438310876138758, 5.934166128374399986265532945999, 6.63976078997539245558031356895, 7.43766285833394858608566111364, 8.05585573692657176565521773082, 8.814909205222430861546432357438, 9.91117382259272941809597850314, 10.37016898224297740417144434421, 11.33526842945092178127096824957, 12.2532661273725816566763275064, 12.63807525240249755772688738759, 13.26144974409320849760420487716, 14.15290546412596611561911209654, 14.53337841028564871369053783838, 15.73271698466518448550929472318, 16.6374136496196167585780488468, 17.20299617543514236066611017086, 17.58339887170905443521720033779, 18.140646331675648438452288364732, 19.30947015937711686501534481075

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