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Properties

Label 1-2624-2624.1331-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

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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: $\chi_{2624} (1331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ -0.935 - 0.352i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2867549303 - 1.575609534i\)
\(L(\frac12)\) \(\approx\) \(0.2867549303 - 1.575609534i\)
\(L(1)\) \(\approx\) \(0.8599346299 - 0.6616967272i\)
\(L(1)\) \(\approx\) \(0.8599346299 - 0.6616967272i\)

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 \( 1 + (0.453 - 0.891i)T \)
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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

−19.82865969051599082351276902426, −18.819594609574750400696732342770, −18.09041753826362942619786891855, −17.5576952250418993661060417924, −16.91680854880952513191441841767, −15.99207309314255626779182921496, −15.281253696179731787533452841860, −15.05339326065295473636282404710, −13.98578203495033512354317704687, −13.360767613220207790941444811229, −12.3860079152906290585661126619, −11.681268423057622860489128428333, −10.91867522532314967041327315270, −10.280287482425840671630832713272, −9.40392474646151258357839060789, −9.23480119614999152754091131647, −8.180818805557827533557896816506, −6.84303901580859996656917930797, −6.34612078775991109865170423364, −5.6567704448941444710496996990, −4.97686308391507522809027992715, −3.80658354149525279835936259292, −3.2830377345040923453031967255, −2.33102371532751038510705260876, −1.308658684523939042976214971, 0.609248347465284725526989635451, 1.15101042633681609806217290916, 2.11134656201363728485209369941, 3.15682171185452134502474734490, 4.07656475021802377079304291209, 5.12885947720118675629702628537, 5.85346637317939967879038021956, 6.59520623202124890721271666714, 6.99888624267236175424074492512, 8.20838695026118084963636858090, 8.735283656128048556249667439506, 9.58168388924584019804427253487, 10.45738313283946860038727841223, 11.16485949600217666329870929846, 12.02278904931880305388493067999, 12.59871957353329563923251046190, 13.60366351696917067819487224094, 13.715735527913616060130417609104, 14.299634147666723226538421449095, 15.92721002050149352444618057704, 16.38100942932064970908897146266, 16.841812665532668909187308098803, 17.71761767458813334463745409947, 18.21402605091117984155572172998, 18.98072804707906147937946183115

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