Properties

Label 2-3675-1.1-c1-0-9
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  + 0.381·2-s − 3-s − 1.85·4-s − 0.381·6-s − 1.47·8-s + 9-s + 3.47·11-s + 1.85·12-s − 5.23·13-s + 3.14·16-s − 5.70·17-s + 0.381·18-s − 1.23·19-s + 1.32·22-s + 5·23-s + 1.47·24-s − 2·26-s − 27-s − 8.70·29-s + 4.47·31-s + 4.14·32-s − 3.47·33-s − 2.18·34-s − 1.85·36-s − 3.47·37-s − 0.472·38-s + 5.23·39-s + ⋯
L(s)  = 1  + 0.270·2-s − 0.577·3-s − 0.927·4-s − 0.155·6-s − 0.520·8-s + 0.333·9-s + 1.04·11-s + 0.535·12-s − 1.45·13-s + 0.786·16-s − 1.38·17-s + 0.0900·18-s − 0.283·19-s + 0.282·22-s + 1.04·23-s + 0.300·24-s − 0.392·26-s − 0.192·27-s − 1.61·29-s + 0.803·31-s + 0.732·32-s − 0.604·33-s − 0.373·34-s − 0.309·36-s − 0.570·37-s − 0.0765·38-s + 0.838·39-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9313898740\)
\(L(\frac12)\) \(\approx\) \(0.9313898740\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.381T + 2T^{2} \)
11 \( 1 - 3.47T + 11T^{2} \)
13 \( 1 + 5.23T + 13T^{2} \)
17 \( 1 + 5.70T + 17T^{2} \)
19 \( 1 + 1.23T + 19T^{2} \)
23 \( 1 - 5T + 23T^{2} \)
29 \( 1 + 8.70T + 29T^{2} \)
31 \( 1 - 4.47T + 31T^{2} \)
37 \( 1 + 3.47T + 37T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 3.76T + 43T^{2} \)
47 \( 1 + 2.76T + 47T^{2} \)
53 \( 1 + 8.47T + 53T^{2} \)
59 \( 1 - 5.23T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 - 10.7T + 67T^{2} \)
71 \( 1 + 9.47T + 71T^{2} \)
73 \( 1 - 3.23T + 73T^{2} \)
79 \( 1 - 6.23T + 79T^{2} \)
83 \( 1 + 3.52T + 83T^{2} \)
89 \( 1 + 7.70T + 89T^{2} \)
97 \( 1 + 3.52T + 97T^{2} \)
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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

−8.747647693850141182889749904064, −7.69395778238630736882848905312, −6.96231548876517594084400217725, −6.24409196518365822927981307405, −5.41071803786967994380833471063, −4.61345258763112039078115469402, −4.24288358413831681207813950677, −3.15173313892149504603661228994, −1.94812799653251250767143109003, −0.55167249114830531759465687399, 0.55167249114830531759465687399, 1.94812799653251250767143109003, 3.15173313892149504603661228994, 4.24288358413831681207813950677, 4.61345258763112039078115469402, 5.41071803786967994380833471063, 6.24409196518365822927981307405, 6.96231548876517594084400217725, 7.69395778238630736882848905312, 8.747647693850141182889749904064

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