Properties

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

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: \(1\)
Selberg data: \((2,\ 3675,\ (\ :1/2),\ -1)\)

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 \( 1 \)
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

−8.147780811848624937742704656157, −7.49481170028502263168271760327, −6.81005579146774916998539383137, −5.65490291255029523785641071670, −5.20990385709902739148000490349, −4.46161838387592170608390615408, −3.59974204500420010922240062161, −2.44105420011153060180056091389, −1.10842947150966580383950259716, 0, 1.10842947150966580383950259716, 2.44105420011153060180056091389, 3.59974204500420010922240062161, 4.46161838387592170608390615408, 5.20990385709902739148000490349, 5.65490291255029523785641071670, 6.81005579146774916998539383137, 7.49481170028502263168271760327, 8.147780811848624937742704656157

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