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Properties

Label 1-2624-2624.1019-r0-0-0
Degree $1$
Conductor $2624$
Sign $-0.890 - 0.455i$
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  + (−0.923 − 0.382i)3-s + (−0.760 − 0.649i)5-s + (−0.809 − 0.587i)7-s + (0.707 + 0.707i)9-s + (−0.760 + 0.649i)11-s + (0.233 + 0.972i)13-s + (0.453 + 0.891i)15-s + (0.891 + 0.453i)17-s + (−0.522 − 0.852i)19-s + (0.522 + 0.852i)21-s + (−0.156 + 0.987i)23-s + (0.156 + 0.987i)25-s + (−0.382 − 0.923i)27-s + (−0.649 + 0.760i)29-s + (−0.309 − 0.951i)31-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)3-s + (−0.760 − 0.649i)5-s + (−0.809 − 0.587i)7-s + (0.707 + 0.707i)9-s + (−0.760 + 0.649i)11-s + (0.233 + 0.972i)13-s + (0.453 + 0.891i)15-s + (0.891 + 0.453i)17-s + (−0.522 − 0.852i)19-s + (0.522 + 0.852i)21-s + (−0.156 + 0.987i)23-s + (0.156 + 0.987i)25-s + (−0.382 − 0.923i)27-s + (−0.649 + 0.760i)29-s + (−0.309 − 0.951i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $-0.890 - 0.455i$
Analytic conductor: \(12.1858\)
Root analytic conductor: \(12.1858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ -0.890 - 0.455i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05251098369 - 0.2179678216i\)
\(L(\frac12)\) \(\approx\) \(0.05251098369 - 0.2179678216i\)
\(L(1)\) \(\approx\) \(0.5324326764 - 0.09648774944i\)
\(L(1)\) \(\approx\) \(0.5324326764 - 0.09648774944i\)

Euler product

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

−19.25009885082452668139728871638, −18.987048360906206778038155497144, −18.25107076151891524617031533019, −17.68296562755043857017195156737, −16.47273373149877743392042553376, −16.17320595765127415858965976410, −15.58338725468457480926202666359, −14.88818607405738322100011195603, −14.107790098230791726645116019782, −12.79511195412955508191855856779, −12.54974317012337723547893696557, −11.74568620692971951271742180999, −10.89592209884835650067830384929, −10.43774913204249548195522699574, −9.80413731212043363003870979553, −8.72719593236101625853181447414, −7.9060866152860828184445622631, −7.16395577833456566461896553064, −6.13285725488734726813554310874, −5.81377466380776193894780021096, −4.92627015247390547083420445055, −3.79710643510774395491487390569, −3.28948109148576597481672945723, −2.43046078047674060322946141129, −0.7831449028006460071921344821, 0.12269599163574295926379443469, 1.19362830498372663257441830667, 2.09770868113729284423072403097, 3.49032699101661357189750672202, 4.164518161950699905376366530666, 4.958942554492843395011872679, 5.68223992415867221677116271690, 6.65335771206087169041973658921, 7.30607916016340049649006128939, 7.80964107052433333770195770305, 8.91257092211313738700670102508, 9.73496342793224407137707316333, 10.47747078924464861619005191049, 11.265278675593683534260893236261, 11.87932164989269499908318253167, 12.71344955039099448612412659462, 13.04001262678259805040519426253, 13.79696799750814786824228560114, 15.01231833967050420147900068860, 15.75920286705831711427695752352, 16.27456665517745828029793853328, 16.95083263107695833242193642844, 17.396889318127934039506944277446, 18.455638199916717340850938735038, 19.091136897713514316248918585309

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