Properties

Label 2-6045-1.1-c1-0-73
Degree $2$
Conductor $6045$
Sign $1$
Analytic cond. $48.2695$
Root an. cond. $6.94763$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s − 4-s − 5-s + 6-s − 3·8-s + 9-s − 10-s + 4·11-s − 12-s + 13-s − 15-s − 16-s + 2·17-s + 18-s − 4·19-s + 20-s + 4·22-s − 3·24-s + 25-s + 26-s + 27-s − 6·29-s − 30-s + 31-s + 5·32-s + 4·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.277·13-s − 0.258·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.852·22-s − 0.612·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 1.11·29-s − 0.182·30-s + 0.179·31-s + 0.883·32-s + 0.696·33-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6045,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.719547584\)
\(L(\frac12)\) \(\approx\) \(2.719547584\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.209710972116466562550546370997, −7.38671189921863030164643513979, −6.50599855618906477641565296043, −5.96418744495456884644347538717, −5.01048576848350296742258145519, −4.22408331721135995325885066726, −3.79135413163933284660599477193, −3.11909050832711916382635306522, −2.00406461229721458192982337891, −0.76844534091880910640983837253, 0.76844534091880910640983837253, 2.00406461229721458192982337891, 3.11909050832711916382635306522, 3.79135413163933284660599477193, 4.22408331721135995325885066726, 5.01048576848350296742258145519, 5.96418744495456884644347538717, 6.50599855618906477641565296043, 7.38671189921863030164643513979, 8.209710972116466562550546370997

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