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Properties

Label 1-2624-2624.133-r0-0-0
Degree $1$
Conductor $2624$
Sign $-0.878 + 0.478i$
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 + (−0.972 + 0.233i)5-s + (−0.891 + 0.453i)7-s + (−0.707 + 0.707i)9-s + (0.852 − 0.522i)11-s + (0.649 + 0.760i)13-s + (−0.587 − 0.809i)15-s + (−0.587 + 0.809i)17-s + (0.0784 + 0.996i)19-s + (−0.760 − 0.649i)21-s + (0.891 + 0.453i)23-s + (0.891 − 0.453i)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 + (−0.972 + 0.233i)5-s + (−0.891 + 0.453i)7-s + (−0.707 + 0.707i)9-s + (0.852 − 0.522i)11-s + (0.649 + 0.760i)13-s + (−0.587 − 0.809i)15-s + (−0.587 + 0.809i)17-s + (0.0784 + 0.996i)19-s + (−0.760 − 0.649i)21-s + (0.891 + 0.453i)23-s + (0.891 − 0.453i)25-s + (−0.923 − 0.382i)27-s + (0.852 + 0.522i)29-s + (0.809 + 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.878 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.878 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $-0.878 + 0.478i$
Analytic conductor: \(12.1858\)
Root analytic conductor: \(12.1858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ -0.878 + 0.478i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3549009953 + 1.394598090i\)
\(L(\frac12)\) \(\approx\) \(0.3549009953 + 1.394598090i\)
\(L(1)\) \(\approx\) \(0.8417831345 + 0.5710767799i\)
\(L(1)\) \(\approx\) \(0.8417831345 + 0.5710767799i\)

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

−19.220492913041023922449514354696, −18.52930070883817485863839888308, −17.57628448954117256267693269663, −17.12824464252497394810988808911, −16.08641483830653373148431456922, −15.51660153035212711830029689860, −14.854586605148016549127107755714, −13.88376204373153527761849258639, −13.24065401729728742275165672138, −12.73163657509544166208659087324, −11.94155834369390369423501205965, −11.36218073067870668990097110744, −10.48605848672073769698208386715, −9.26086577661480235053129112520, −8.967636172231015076005589408317, −7.947042972309963122344160137186, −7.328765799372130228443764673, −6.682571915303847067455075868322, −6.0887515552927920311631648472, −4.7103798632708443079091539870, −4.04877134526940165914357501582, −3.06516410824825833185001086426, −2.579609263737330990265625394665, −1.01952616489978394690673990852, −0.590974860600639537168590816279, 1.14621946381512173931927629478, 2.52144058677549183586279557320, 3.32607312330337328827320105473, 3.88745780025306265030200359242, 4.44842610626345663611072734604, 5.67262601861616509080082868285, 6.378955104036654321546437174969, 7.15037285197043837546885775586, 8.38503375857042107313905478814, 8.67691452351304060527397847442, 9.416320937111589884400279559428, 10.291374462303672811785204753997, 11.02291904928437326992406095343, 11.63516280893489458398514062060, 12.37883031597120165829902885881, 13.30486141524182834620085489278, 14.19314156968423321442351178561, 14.72691273844087152308124837426, 15.53488996620924593828866884961, 16.03581337218490406606391627267, 16.524819826257799709991166937345, 17.30931277306987001855206903784, 18.52422357726993541513504256923, 19.23039925115312878246257252445, 19.503027005968744483677949434448

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