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Properties

Label 1-2624-2624.685-r1-0-0
Degree $1$
Conductor $2624$
Sign $-0.293 + 0.955i$
Analytic cond. $281.987$
Root an. cond. $281.987$
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 + (−0.233 + 0.972i)5-s + (0.309 + 0.951i)7-s + (0.707 − 0.707i)9-s + (−0.233 − 0.972i)11-s + (0.996 − 0.0784i)13-s + (0.156 + 0.987i)15-s + (0.987 + 0.156i)17-s + (−0.649 − 0.760i)19-s + (0.649 + 0.760i)21-s + (−0.891 + 0.453i)23-s + (−0.891 − 0.453i)25-s + (0.382 − 0.923i)27-s + (0.972 + 0.233i)29-s + (−0.809 + 0.587i)31-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)3-s + (−0.233 + 0.972i)5-s + (0.309 + 0.951i)7-s + (0.707 − 0.707i)9-s + (−0.233 − 0.972i)11-s + (0.996 − 0.0784i)13-s + (0.156 + 0.987i)15-s + (0.987 + 0.156i)17-s + (−0.649 − 0.760i)19-s + (0.649 + 0.760i)21-s + (−0.891 + 0.453i)23-s + (−0.891 − 0.453i)25-s + (0.382 − 0.923i)27-s + (0.972 + 0.233i)29-s + (−0.809 + 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.293 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.293 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $-0.293 + 0.955i$
Analytic conductor: \(281.987\)
Root analytic conductor: \(281.987\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (1:\ ),\ -0.293 + 0.955i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.512050650 + 2.045840126i\)
\(L(\frac12)\) \(\approx\) \(1.512050650 + 2.045840126i\)
\(L(1)\) \(\approx\) \(1.400224631 + 0.2459824947i\)
\(L(1)\) \(\approx\) \(1.400224631 + 0.2459824947i\)

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

−19.10900449835845687953407560340, −18.34982855634941309473285328701, −17.45909874004404502324799979245, −16.6062689743602770285200502181, −16.2261400020767807704931670317, −15.40523817195914833522628369434, −14.69238216644676048153574997534, −13.9061783475164193981269281509, −13.432650969685625319648951708, −12.55706474495554889532931627415, −12.00364373372822847122633804598, −10.77543649230463830108267229920, −10.20319400975773510048367090073, −9.57215473254557062838597349988, −8.64703613582359014268732540831, −8.04404759561279190355232086679, −7.59155190101108168477256621267, −6.561069054376756012118944131323, −5.37923552162064937726190937771, −4.602953729412677391120477872872, −3.97025016373367355431502381584, −3.43661474135581110337190989014, −1.99173982976457620876056274946, −1.50223326202085006578667494833, −0.34679260743822183642661619885, 1.08230892728358661778987679778, 1.99954073762277834458764684330, 2.91872773760686552164346084250, 3.312654052743139280198266646037, 4.21387854056393267501178478555, 5.56039322043581936093084446482, 6.16548897745166463440966490377, 6.96680370817278457535667803057, 7.853571085981269122769290154912, 8.4623773327349323843090239371, 8.93787517094444770909734779509, 10.02606502529765451542008641360, 10.723399317530304011142892244543, 11.55198549305716346017345713299, 12.17678368455127162645608557843, 13.13826251852709587008567278605, 13.7524903906710606751238771128, 14.54052630134938789445750392309, 14.87530412454841473555424520065, 15.9024903056267596323797845686, 16.09324372411352923742462197271, 17.76970064665563757464024152959, 18.029181088281118920604096704667, 18.8134710252256181337443539200, 19.24726266837049033652802444593

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