Properties

Label 2-6045-1.1-c1-0-50
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·2-s − 3-s + 0.999·4-s − 5-s − 1.73·6-s + 2·7-s − 1.73·8-s + 9-s − 1.73·10-s − 5.73·11-s − 0.999·12-s + 13-s + 3.46·14-s + 15-s − 5·16-s + 3.46·17-s + 1.73·18-s + 3·19-s − 0.999·20-s − 2·21-s − 9.92·22-s − 3·23-s + 1.73·24-s + 25-s + 1.73·26-s − 27-s + 1.99·28-s + ⋯
L(s)  = 1  + 1.22·2-s − 0.577·3-s + 0.499·4-s − 0.447·5-s − 0.707·6-s + 0.755·7-s − 0.612·8-s + 0.333·9-s − 0.547·10-s − 1.72·11-s − 0.288·12-s + 0.277·13-s + 0.925·14-s + 0.258·15-s − 1.25·16-s + 0.840·17-s + 0.408·18-s + 0.688·19-s − 0.223·20-s − 0.436·21-s − 2.11·22-s − 0.625·23-s + 0.353·24-s + 0.200·25-s + 0.339·26-s − 0.192·27-s + 0.377·28-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: \(0\)
Selberg data: \((2,\ 6045,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.230664690\)
\(L(\frac12)\) \(\approx\) \(2.230664690\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T + 2T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + 5.73T + 11T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 - 3T + 19T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 + 1.73T + 29T^{2} \)
37 \( 1 - 4.92T + 37T^{2} \)
41 \( 1 - T + 41T^{2} \)
43 \( 1 - 4.53T + 43T^{2} \)
47 \( 1 + 1.73T + 47T^{2} \)
53 \( 1 - 7.46T + 53T^{2} \)
59 \( 1 + 0.535T + 59T^{2} \)
61 \( 1 + 5.46T + 61T^{2} \)
67 \( 1 - 10T + 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 2.53T + 73T^{2} \)
79 \( 1 - 3.46T + 79T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + 6.92T + 97T^{2} \)
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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

−7.79485966493280057200044721556, −7.46020549276098321671806146249, −6.33383734213906069284085341950, −5.63290647059910425010505665298, −5.18409154894500431033114491655, −4.62226446566986656710356120173, −3.79580255759999628935158958985, −3.03491672170944997027613150799, −2.10811753533648679160773344294, −0.65454203330545635071707779697, 0.65454203330545635071707779697, 2.10811753533648679160773344294, 3.03491672170944997027613150799, 3.79580255759999628935158958985, 4.62226446566986656710356120173, 5.18409154894500431033114491655, 5.63290647059910425010505665298, 6.33383734213906069284085341950, 7.46020549276098321671806146249, 7.79485966493280057200044721556

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