Properties

Label 2-2415-1.1-c1-0-43
Degree $2$
Conductor $2415$
Sign $1$
Analytic cond. $19.2838$
Root an. cond. $4.39134$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.43·2-s + 3-s + 0.0581·4-s − 5-s + 1.43·6-s − 7-s − 2.78·8-s + 9-s − 1.43·10-s + 4.68·11-s + 0.0581·12-s − 0.389·13-s − 1.43·14-s − 15-s − 4.11·16-s + 6.87·17-s + 1.43·18-s − 2.86·19-s − 0.0581·20-s − 21-s + 6.72·22-s − 23-s − 2.78·24-s + 25-s − 0.559·26-s + 27-s − 0.0581·28-s + ⋯
L(s)  = 1  + 1.01·2-s + 0.577·3-s + 0.0290·4-s − 0.447·5-s + 0.585·6-s − 0.377·7-s − 0.984·8-s + 0.333·9-s − 0.453·10-s + 1.41·11-s + 0.0167·12-s − 0.108·13-s − 0.383·14-s − 0.258·15-s − 1.02·16-s + 1.66·17-s + 0.338·18-s − 0.657·19-s − 0.0129·20-s − 0.218·21-s + 1.43·22-s − 0.208·23-s − 0.568·24-s + 0.200·25-s − 0.109·26-s + 0.192·27-s − 0.0109·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(19.2838\)
Root analytic conductor: \(4.39134\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2415,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.183719992\)
\(L(\frac12)\) \(\approx\) \(3.183719992\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 + T \)
good2 \( 1 - 1.43T + 2T^{2} \)
11 \( 1 - 4.68T + 11T^{2} \)
13 \( 1 + 0.389T + 13T^{2} \)
17 \( 1 - 6.87T + 17T^{2} \)
19 \( 1 + 2.86T + 19T^{2} \)
29 \( 1 - 7.41T + 29T^{2} \)
31 \( 1 - 5.33T + 31T^{2} \)
37 \( 1 - 1.15T + 37T^{2} \)
41 \( 1 + 1.56T + 41T^{2} \)
43 \( 1 + 9.26T + 43T^{2} \)
47 \( 1 - 7.01T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 - 13.2T + 59T^{2} \)
61 \( 1 - 2.54T + 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 + 0.362T + 73T^{2} \)
79 \( 1 + 16.9T + 79T^{2} \)
83 \( 1 + 5.09T + 83T^{2} \)
89 \( 1 - 3.68T + 89T^{2} \)
97 \( 1 - 8.66T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.672797028109315503283166877922, −8.468555585419555234095881839073, −7.24544501717296217633031442442, −6.53946863348498908726829522121, −5.78678167139453600878221534037, −4.77973920134898576626576975805, −3.98575923987132547022381477371, −3.48341166647645092936296591472, −2.58851883734769754947546798274, −1.00328553483542726692762426298, 1.00328553483542726692762426298, 2.58851883734769754947546798274, 3.48341166647645092936296591472, 3.98575923987132547022381477371, 4.77973920134898576626576975805, 5.78678167139453600878221534037, 6.53946863348498908726829522121, 7.24544501717296217633031442442, 8.468555585419555234095881839073, 8.672797028109315503283166877922

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