Properties

Label 2-2415-1.1-c1-0-45
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.92·2-s − 3-s + 1.70·4-s − 5-s + 1.92·6-s − 7-s + 0.576·8-s + 9-s + 1.92·10-s − 5.42·11-s − 1.70·12-s + 2.22·13-s + 1.92·14-s + 15-s − 4.50·16-s − 2·17-s − 1.92·18-s + 6.85·19-s − 1.70·20-s + 21-s + 10.4·22-s − 23-s − 0.576·24-s + 25-s − 4.27·26-s − 27-s − 1.70·28-s + ⋯
L(s)  = 1  − 1.36·2-s − 0.577·3-s + 0.850·4-s − 0.447·5-s + 0.785·6-s − 0.377·7-s + 0.203·8-s + 0.333·9-s + 0.608·10-s − 1.63·11-s − 0.490·12-s + 0.616·13-s + 0.514·14-s + 0.258·15-s − 1.12·16-s − 0.485·17-s − 0.453·18-s + 1.57·19-s − 0.380·20-s + 0.218·21-s + 2.22·22-s − 0.208·23-s − 0.117·24-s + 0.200·25-s − 0.838·26-s − 0.192·27-s − 0.321·28-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: \(1\)
Selberg data: \((2,\ 2415,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.92T + 2T^{2} \)
11 \( 1 + 5.42T + 11T^{2} \)
13 \( 1 - 2.22T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 6.85T + 19T^{2} \)
29 \( 1 - 4.49T + 29T^{2} \)
31 \( 1 - 0.513T + 31T^{2} \)
37 \( 1 + 2.87T + 37T^{2} \)
41 \( 1 + 0.509T + 41T^{2} \)
43 \( 1 + 3.71T + 43T^{2} \)
47 \( 1 - 1.20T + 47T^{2} \)
53 \( 1 + 8.82T + 53T^{2} \)
59 \( 1 - 8.21T + 59T^{2} \)
61 \( 1 - 2.25T + 61T^{2} \)
67 \( 1 - 6.40T + 67T^{2} \)
71 \( 1 - 14.3T + 71T^{2} \)
73 \( 1 + 1.48T + 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 + 3.80T + 83T^{2} \)
89 \( 1 + 14.2T + 89T^{2} \)
97 \( 1 - 8.36T + 97T^{2} \)
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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

−8.422906962517877834126632856019, −8.000515332010103752468841305645, −7.23064809501071904441687308271, −6.55500283724255144652814165170, −5.43363483300080064224737261016, −4.76874164883960347585013073173, −3.52892854823574043520495298055, −2.41912480535167910338672323361, −1.05079795616744285391470580377, 0, 1.05079795616744285391470580377, 2.41912480535167910338672323361, 3.52892854823574043520495298055, 4.76874164883960347585013073173, 5.43363483300080064224737261016, 6.55500283724255144652814165170, 7.23064809501071904441687308271, 8.000515332010103752468841305645, 8.422906962517877834126632856019

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