Properties

Degree $2$
Conductor $3675$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.414·2-s + 3-s − 1.82·4-s − 0.414·6-s + 1.58·8-s + 9-s − 2·11-s − 1.82·12-s + 2.58·13-s + 3·16-s − 2.24·17-s − 0.414·18-s + 2.82·19-s + 0.828·22-s + 7.65·23-s + 1.58·24-s − 1.07·26-s + 27-s − 6.82·29-s + 1.17·31-s − 4.41·32-s − 2·33-s + 0.928·34-s − 1.82·36-s + 4·37-s − 1.17·38-s + 2.58·39-s + ⋯
L(s)  = 1  − 0.292·2-s + 0.577·3-s − 0.914·4-s − 0.169·6-s + 0.560·8-s + 0.333·9-s − 0.603·11-s − 0.527·12-s + 0.717·13-s + 0.750·16-s − 0.543·17-s − 0.0976·18-s + 0.648·19-s + 0.176·22-s + 1.59·23-s + 0.323·24-s − 0.210·26-s + 0.192·27-s − 1.26·29-s + 0.210·31-s − 0.780·32-s − 0.348·33-s + 0.159·34-s − 0.304·36-s + 0.657·37-s − 0.190·38-s + 0.414·39-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: $1$
Motivic weight: \(1\)
Character: $\chi_{3675} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.539530862\)
\(L(\frac12)\) \(\approx\) \(1.539530862\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.414T + 2T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 2.58T + 13T^{2} \)
17 \( 1 + 2.24T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 - 7.65T + 23T^{2} \)
29 \( 1 + 6.82T + 29T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 1.17T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 - 5.65T + 67T^{2} \)
71 \( 1 - 9.31T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 - 7.31T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 - 2.58T + 97T^{2} \)
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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

−8.518941975047728879078266854356, −7.999832316299798033765249200153, −7.27807335604176674189236749169, −6.39824163937052954811198460182, −5.27697974758996349470661069181, −4.82051074060462128697470921502, −3.75349692209605763541666174657, −3.15740738618990452251776045459, −1.90980432137990327391261256039, −0.75808932091979279254199746409, 0.75808932091979279254199746409, 1.90980432137990327391261256039, 3.15740738618990452251776045459, 3.75349692209605763541666174657, 4.82051074060462128697470921502, 5.27697974758996349470661069181, 6.39824163937052954811198460182, 7.27807335604176674189236749169, 7.999832316299798033765249200153, 8.518941975047728879078266854356

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