Properties

Label 2-3675-1.1-c1-0-32
Degree $2$
Conductor $3675$
Sign $1$
Analytic cond. $29.3450$
Root an. cond. $5.41710$
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 + 6-s − 3·8-s + 9-s − 6·11-s − 12-s − 2·13-s − 16-s + 4·17-s + 18-s + 6·19-s − 6·22-s − 3·24-s − 2·26-s + 27-s − 2·29-s + 10·31-s + 5·32-s − 6·33-s + 4·34-s − 36-s + 4·37-s + 6·38-s − 2·39-s − 2·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 1.80·11-s − 0.288·12-s − 0.554·13-s − 1/4·16-s + 0.970·17-s + 0.235·18-s + 1.37·19-s − 1.27·22-s − 0.612·24-s − 0.392·26-s + 0.192·27-s − 0.371·29-s + 1.79·31-s + 0.883·32-s − 1.04·33-s + 0.685·34-s − 1/6·36-s + 0.657·37-s + 0.973·38-s − 0.320·39-s − 0.312·41-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: yes
Arithmetic: yes
Character: $\chi_{3675} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.355304227\)
\(L(\frac12)\) \(\approx\) \(2.355304227\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
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

−8.245873608153951043349925317243, −7.976741612740113433163431746666, −7.18600650759489131977738022987, −6.06490562142255136482234133151, −5.20476390495317100873412386991, −4.93935473109955635441904702605, −3.83469354318778825175338832285, −3.01958613902430109855645796176, −2.47409506150476348759674637901, −0.77490080633513895615658874599, 0.77490080633513895615658874599, 2.47409506150476348759674637901, 3.01958613902430109855645796176, 3.83469354318778825175338832285, 4.93935473109955635441904702605, 5.20476390495317100873412386991, 6.06490562142255136482234133151, 7.18600650759489131977738022987, 7.976741612740113433163431746666, 8.245873608153951043349925317243

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