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Properties

Label 1-2624-2624.1283-r0-0-0
Degree $1$
Conductor $2624$
Sign $0.0108 - 0.999i$
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 + (−0.309 − 0.951i)7-s + (−0.707 + 0.707i)9-s + (−0.972 + 0.233i)11-s + (−0.0784 − 0.996i)13-s + (0.156 + 0.987i)15-s + (−0.987 − 0.156i)17-s + (0.760 − 0.649i)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 + (0.233 − 0.972i)29-s + (0.809 − 0.587i)31-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)3-s + (0.972 + 0.233i)5-s + (−0.309 − 0.951i)7-s + (−0.707 + 0.707i)9-s + (−0.972 + 0.233i)11-s + (−0.0784 − 0.996i)13-s + (0.156 + 0.987i)15-s + (−0.987 − 0.156i)17-s + (0.760 − 0.649i)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 + (0.233 − 0.972i)29-s + (0.809 − 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0108 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0108 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $0.0108 - 0.999i$
Analytic conductor: \(12.1858\)
Root analytic conductor: \(12.1858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (1283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ 0.0108 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6403190740 - 0.6472862894i\)
\(L(\frac12)\) \(\approx\) \(0.6403190740 - 0.6472862894i\)
\(L(1)\) \(\approx\) \(1.040146810 + 0.1001792252i\)
\(L(1)\) \(\approx\) \(1.040146810 + 0.1001792252i\)

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

−19.51465869948514151425425820910, −18.57552207214776886353401970554, −18.19403969415225237300567199324, −17.78203957088264996058276426845, −16.65500331151655450352588900708, −16.09340703572074264536704734786, −15.19777431368704068728386040129, −14.28937929542459854688092047206, −13.802511530605764455666509787525, −13.14876412797175598603280054177, −12.42238352094140633209207951487, −11.950665171187041614342447685621, −10.92812730571084997985340478046, −9.96537862749846998825140369504, −9.25503954822980394283223062273, −8.60176501926611959303336294685, −8.04307238777675025283412423826, −6.820783721464610513244155745554, −6.41934009936221210763193252395, −5.57451913266226545569773018312, −4.94091365919206216716288968203, −3.57197666631129681805150913753, −2.586543880244328142980636290139, −2.135854828150922173534923523852, −1.333489840114606102590964477987, 0.24131596649342634788810409481, 1.73318252871845602977769561007, 2.803489915291713325886237574373, 3.13382759439912175447009457444, 4.38365790706836642124806640427, 4.922459431278723151339442993686, 5.79215875607489855568348240534, 6.575770714519237302213491426219, 7.65532722305323673014335889457, 8.15662603835470667334578670025, 9.360842512005433501397960985621, 9.77887238396462861593549646461, 10.45136268002210561113238582414, 10.852002411650688538952555548367, 11.8961020117237985100390778793, 13.24945476689693798182493062700, 13.475295591098396944032689594075, 13.98430818240652779091516640870, 15.21263649439736365118885219273, 15.41685250564671585328301181508, 16.30649242159250403171638666155, 17.08949356665164296935893859823, 17.69073777709499148016684976276, 18.24968179042674467597141834037, 19.39851539051691992846455048613

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