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Properties

Label 1-2624-2624.1109-r0-0-0
Degree $1$
Conductor $2624$
Sign $-0.848 - 0.528i$
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 + (−0.987 + 0.156i)7-s + (−0.707 + 0.707i)9-s + (0.996 + 0.0784i)11-s + (−0.522 + 0.852i)13-s + (−0.309 + 0.951i)15-s + (0.309 + 0.951i)17-s + (−0.972 − 0.233i)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 + (−0.996 + 0.0784i)29-s + (−0.309 − 0.951i)31-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)3-s + (0.760 + 0.649i)5-s + (−0.987 + 0.156i)7-s + (−0.707 + 0.707i)9-s + (0.996 + 0.0784i)11-s + (−0.522 + 0.852i)13-s + (−0.309 + 0.951i)15-s + (0.309 + 0.951i)17-s + (−0.972 − 0.233i)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 + (−0.996 + 0.0784i)29-s + (−0.309 − 0.951i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.848 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.848 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $-0.848 - 0.528i$
Analytic conductor: \(12.1858\)
Root analytic conductor: \(12.1858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ -0.848 - 0.528i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2762497331 + 0.9656173774i\)
\(L(\frac12)\) \(\approx\) \(-0.2762497331 + 0.9656173774i\)
\(L(1)\) \(\approx\) \(0.8427813104 + 0.6091871551i\)
\(L(1)\) \(\approx\) \(0.8427813104 + 0.6091871551i\)

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

−19.03034601024826831483987637684, −18.19616001317141043273388917693, −17.53963441695306958215958407858, −16.86263438367606229379586108850, −16.385792648411167718858308978681, −15.23176584012350433309191749834, −14.54170276244686744388505687399, −13.72959146274037279236776981591, −13.256850562993119020413831045446, −12.53353650675520871210261468590, −12.139977025298279216569318464852, −11.094617185203853459476793072443, −9.98759462745941838240642217426, −9.3747217615284805081447541173, −8.911586100766554121469824867027, −7.91547887703837663526394396428, −7.15694034071779616926720559020, −6.408619569011753196542671590045, −5.80606712291941604628361946390, −4.96054660153089940943383428272, −3.69227012201156209920743296105, −3.04298202818341168417064860601, −2.06736597048080367883591654763, −1.28739342490952072314004523635, −0.27701003540491715732690663431, 1.72427119999067385533243429898, 2.44502913538761133915250959223, 3.28971881772055986466952865354, 4.01619660140438326766703626763, 4.76357672801740755154243508158, 6.014648605269467567570733987925, 6.31595681228168494173656379354, 7.186336891786274019052895488359, 8.359931681024736987694846831781, 9.186875702631860137791871027589, 9.59703477893418987238420335413, 10.23047590705344304332925779032, 10.97314621996300596264702007647, 11.73692680829706395411092048909, 12.818278667969662873217897706635, 13.37649535509260268414566984355, 14.380693416269926514254248455271, 14.769837184062329503902566405583, 15.242846633518312473803476073106, 16.56845367402106013605294374294, 16.6963917304550586447597746444, 17.375438898473161927410532227878, 18.598339923074685860587687517206, 19.12634773304802523106656086321, 19.68581843770782720111520040992

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