Properties

Label 1-2415-2415.614-r0-0-0
Degree $1$
Conductor $2415$
Sign $-0.419 + 0.907i$
Analytic cond. $11.2152$
Root an. cond. $11.2152$
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  + (−0.786 + 0.618i)2-s + (0.235 − 0.971i)4-s + (0.415 + 0.909i)8-s + (0.786 + 0.618i)11-s + (0.841 + 0.540i)13-s + (−0.888 − 0.458i)16-s + (−0.723 + 0.690i)17-s + (−0.723 − 0.690i)19-s − 22-s + (−0.995 + 0.0950i)26-s + (0.959 + 0.281i)29-s + (0.995 + 0.0950i)31-s + (0.981 − 0.189i)32-s + (0.142 − 0.989i)34-s + (0.327 − 0.945i)37-s + (0.995 + 0.0950i)38-s + ⋯
L(s)  = 1  + (−0.786 + 0.618i)2-s + (0.235 − 0.971i)4-s + (0.415 + 0.909i)8-s + (0.786 + 0.618i)11-s + (0.841 + 0.540i)13-s + (−0.888 − 0.458i)16-s + (−0.723 + 0.690i)17-s + (−0.723 − 0.690i)19-s − 22-s + (−0.995 + 0.0950i)26-s + (0.959 + 0.281i)29-s + (0.995 + 0.0950i)31-s + (0.981 − 0.189i)32-s + (0.142 − 0.989i)34-s + (0.327 − 0.945i)37-s + (0.995 + 0.0950i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $-0.419 + 0.907i$
Analytic conductor: \(11.2152\)
Root analytic conductor: \(11.2152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2415} (614, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2415,\ (0:\ ),\ -0.419 + 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5395750608 + 0.8435521856i\)
\(L(\frac12)\) \(\approx\) \(0.5395750608 + 0.8435521856i\)
\(L(1)\) \(\approx\) \(0.7128604794 + 0.3029296849i\)
\(L(1)\) \(\approx\) \(0.7128604794 + 0.3029296849i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.786 + 0.618i)T \)
11 \( 1 + (0.786 + 0.618i)T \)
13 \( 1 + (0.841 + 0.540i)T \)
17 \( 1 + (-0.723 + 0.690i)T \)
19 \( 1 + (-0.723 - 0.690i)T \)
29 \( 1 + (0.959 + 0.281i)T \)
31 \( 1 + (0.995 + 0.0950i)T \)
37 \( 1 + (0.327 - 0.945i)T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.0475 + 0.998i)T \)
59 \( 1 + (-0.888 + 0.458i)T \)
61 \( 1 + (-0.580 + 0.814i)T \)
67 \( 1 + (-0.928 - 0.371i)T \)
71 \( 1 + (0.142 + 0.989i)T \)
73 \( 1 + (0.235 - 0.971i)T \)
79 \( 1 + (0.0475 - 0.998i)T \)
83 \( 1 + (0.654 + 0.755i)T \)
89 \( 1 + (-0.995 + 0.0950i)T \)
97 \( 1 + (-0.654 + 0.755i)T \)
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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

−19.28077953608519190288705807024, −18.59891195184903224016885045846, −18.04717219752376002024348052293, −17.125585092941904355141744159349, −16.7694296204530128993916906046, −15.7664830683311558551984225338, −15.324983569904715090460516022896, −13.99129889800960233658199813746, −13.540631834470810632384832566534, −12.64834837108075554709677907276, −11.828319807986983047316257839746, −11.355182706600388030738577746632, −10.46685406442621110247845125874, −9.96952861098026731477549013090, −8.84810323044572393609118477341, −8.56627444317707324783359258803, −7.74863422564845699655361605377, −6.64674691423960529330243914409, −6.2267857713059861164647883347, −4.89597781291358360584656238887, −3.929433626825191100671032078, −3.29284789674775794872967706766, −2.35620265764704800283441775324, −1.405414171347568199829755398480, −0.48633478888263603941437266257, 1.09907494694265738760840778450, 1.8093647176059078702447580890, 2.838592446337491638739767811951, 4.28596249712226511135202624002, 4.64381487512330824203581394653, 6.0593458371832063643793837512, 6.40885238591100038808530559188, 7.13704585992972455967762961598, 8.09107174081738156646272557240, 8.835051990228318940887991219344, 9.2643062371133973872872672001, 10.24361170872662812758538260186, 10.89741226488194318450016118288, 11.58798387408601015454550948645, 12.50509661225991843300268489391, 13.5146505447816516512198825134, 14.12177321751384833550145451253, 15.0469040595616204529218498781, 15.411264256401199158910351186433, 16.32553882264460270233649116454, 16.91795714699690777751690897115, 17.707572937543892466571709898264, 18.06164114096827012002813604769, 19.10958798477201629997582463587, 19.58610161231183508611321565068

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