Properties

Label 2-6045-1.1-c1-0-193
Degree $2$
Conductor $6045$
Sign $-1$
Analytic cond. $48.2695$
Root an. cond. $6.94763$
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  − 1.31·2-s + 3-s − 0.265·4-s + 5-s − 1.31·6-s + 1.83·7-s + 2.98·8-s + 9-s − 1.31·10-s − 3.89·11-s − 0.265·12-s + 13-s − 2.41·14-s + 15-s − 3.39·16-s − 5.72·17-s − 1.31·18-s + 3.17·19-s − 0.265·20-s + 1.83·21-s + 5.12·22-s + 4.42·23-s + 2.98·24-s + 25-s − 1.31·26-s + 27-s − 0.487·28-s + ⋯
L(s)  = 1  − 0.931·2-s + 0.577·3-s − 0.132·4-s + 0.447·5-s − 0.537·6-s + 0.692·7-s + 1.05·8-s + 0.333·9-s − 0.416·10-s − 1.17·11-s − 0.0766·12-s + 0.277·13-s − 0.645·14-s + 0.258·15-s − 0.849·16-s − 1.38·17-s − 0.310·18-s + 0.727·19-s − 0.0594·20-s + 0.400·21-s + 1.09·22-s + 0.923·23-s + 0.609·24-s + 0.200·25-s − 0.258·26-s + 0.192·27-s − 0.0920·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6045\)    =    \(3 \cdot 5 \cdot 13 \cdot 31\)
Sign: $-1$
Analytic conductor: \(48.2695\)
Root analytic conductor: \(6.94763\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6045,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
13 \( 1 - T \)
31 \( 1 - T \)
good2 \( 1 + 1.31T + 2T^{2} \)
7 \( 1 - 1.83T + 7T^{2} \)
11 \( 1 + 3.89T + 11T^{2} \)
17 \( 1 + 5.72T + 17T^{2} \)
19 \( 1 - 3.17T + 19T^{2} \)
23 \( 1 - 4.42T + 23T^{2} \)
29 \( 1 + 7.39T + 29T^{2} \)
37 \( 1 + 4.72T + 37T^{2} \)
41 \( 1 + 7.37T + 41T^{2} \)
43 \( 1 - 3.88T + 43T^{2} \)
47 \( 1 - 7.26T + 47T^{2} \)
53 \( 1 + 3.54T + 53T^{2} \)
59 \( 1 + 7.51T + 59T^{2} \)
61 \( 1 + 7.62T + 61T^{2} \)
67 \( 1 + 12.3T + 67T^{2} \)
71 \( 1 + 0.175T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 - 5.79T + 79T^{2} \)
83 \( 1 - 4.95T + 83T^{2} \)
89 \( 1 - 0.109T + 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−7.84788819375662747879968573180, −7.37365392525110414060946048733, −6.55746167473826154785832268117, −5.36401895479931234933903931891, −4.92742204722006975973282750550, −4.08444509208124542595434624048, −3.00085759467330370623096853054, −2.07937277850727288801110376985, −1.36569949871766833189691658110, 0, 1.36569949871766833189691658110, 2.07937277850727288801110376985, 3.00085759467330370623096853054, 4.08444509208124542595434624048, 4.92742204722006975973282750550, 5.36401895479931234933903931891, 6.55746167473826154785832268117, 7.37365392525110414060946048733, 7.84788819375662747879968573180

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