Properties

Label 2-3675-1.1-c1-0-11
Degree $2$
Conductor $3675$
Sign $1$
Analytic cond. $29.3450$
Root an. cond. $5.41710$
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  − 0.732·2-s + 3-s − 1.46·4-s − 0.732·6-s + 2.53·8-s + 9-s − 2.73·11-s − 1.46·12-s − 5.73·13-s + 1.07·16-s − 6.73·17-s − 0.732·18-s − 2.46·19-s + 2·22-s + 1.26·23-s + 2.53·24-s + 4.19·26-s + 27-s + 6.19·29-s + 6.46·31-s − 5.85·32-s − 2.73·33-s + 4.92·34-s − 1.46·36-s − 7.19·37-s + 1.80·38-s − 5.73·39-s + ⋯
L(s)  = 1  − 0.517·2-s + 0.577·3-s − 0.732·4-s − 0.298·6-s + 0.896·8-s + 0.333·9-s − 0.823·11-s − 0.422·12-s − 1.58·13-s + 0.267·16-s − 1.63·17-s − 0.172·18-s − 0.565·19-s + 0.426·22-s + 0.264·23-s + 0.517·24-s + 0.822·26-s + 0.192·27-s + 1.15·29-s + 1.16·31-s − 1.03·32-s − 0.475·33-s + 0.845·34-s − 0.244·36-s − 1.18·37-s + 0.292·38-s − 0.917·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(29.3450\)
Root analytic conductor: \(5.41710\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9242478984\)
\(L(\frac12)\) \(\approx\) \(0.9242478984\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good2 \( 1 + 0.732T + 2T^{2} \)
11 \( 1 + 2.73T + 11T^{2} \)
13 \( 1 + 5.73T + 13T^{2} \)
17 \( 1 + 6.73T + 17T^{2} \)
19 \( 1 + 2.46T + 19T^{2} \)
23 \( 1 - 1.26T + 23T^{2} \)
29 \( 1 - 6.19T + 29T^{2} \)
31 \( 1 - 6.46T + 31T^{2} \)
37 \( 1 + 7.19T + 37T^{2} \)
41 \( 1 - 2.73T + 41T^{2} \)
43 \( 1 - 7.19T + 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 - 8.39T + 53T^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 + 2.66T + 67T^{2} \)
71 \( 1 + 4.19T + 71T^{2} \)
73 \( 1 - 4.66T + 73T^{2} \)
79 \( 1 - 13.3T + 79T^{2} \)
83 \( 1 - 9.12T + 83T^{2} \)
89 \( 1 + 9.12T + 89T^{2} \)
97 \( 1 + 1.07T + 97T^{2} \)
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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

−8.608108318057106231723480112072, −7.976457120449461484012366711329, −7.24496926290771719885296476497, −6.57575023916773580080017121807, −5.25750341646955001779567090030, −4.69110806290212118607507212310, −4.04961658185705659145801395478, −2.72982611470807994960052462554, −2.13662481612679374389725764019, −0.57018712643604548490477586408, 0.57018712643604548490477586408, 2.13662481612679374389725764019, 2.72982611470807994960052462554, 4.04961658185705659145801395478, 4.69110806290212118607507212310, 5.25750341646955001779567090030, 6.57575023916773580080017121807, 7.24496926290771719885296476497, 7.976457120449461484012366711329, 8.608108318057106231723480112072

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