Properties

Label 2-3675-1.1-c1-0-24
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  − 0.618·2-s − 3-s − 1.61·4-s + 0.618·6-s + 2.23·8-s + 9-s + 2.23·11-s + 1.61·12-s − 2.47·13-s + 1.85·16-s + 6.47·17-s − 0.618·18-s + 6.47·19-s − 1.38·22-s − 0.236·23-s − 2.23·24-s + 1.52·26-s − 27-s − 3·29-s + 4·31-s − 5.61·32-s − 2.23·33-s − 4.00·34-s − 1.61·36-s + 5.47·37-s − 4.00·38-s + 2.47·39-s + ⋯
L(s)  = 1  − 0.437·2-s − 0.577·3-s − 0.809·4-s + 0.252·6-s + 0.790·8-s + 0.333·9-s + 0.674·11-s + 0.467·12-s − 0.685·13-s + 0.463·16-s + 1.56·17-s − 0.145·18-s + 1.48·19-s − 0.294·22-s − 0.0492·23-s − 0.456·24-s + 0.299·26-s − 0.192·27-s − 0.557·29-s + 0.718·31-s − 0.993·32-s − 0.389·33-s − 0.685·34-s − 0.269·36-s + 0.899·37-s − 0.648·38-s + 0.395·39-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

\(L(1)\) \(\approx\) \(1.025346468\)
\(L(\frac12)\) \(\approx\) \(1.025346468\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.618T + 2T^{2} \)
11 \( 1 - 2.23T + 11T^{2} \)
13 \( 1 + 2.47T + 13T^{2} \)
17 \( 1 - 6.47T + 17T^{2} \)
19 \( 1 - 6.47T + 19T^{2} \)
23 \( 1 + 0.236T + 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 5.47T + 37T^{2} \)
41 \( 1 + 8.94T + 41T^{2} \)
43 \( 1 + 8.70T + 43T^{2} \)
47 \( 1 - 1.52T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 2.47T + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 - 0.708T + 67T^{2} \)
71 \( 1 - 3.76T + 71T^{2} \)
73 \( 1 + 5.52T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 0.944T + 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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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

−8.501951421340583370945141673758, −7.80032570571314572026789764339, −7.23713402589484232943368699180, −6.29083246312614561869923438872, −5.30852168681852901648024062341, −4.98624811114539665539354289203, −3.91037351924812594223985792086, −3.16537886852318153350254269490, −1.57050684715428305011054075461, −0.71075795450917434645347341932, 0.71075795450917434645347341932, 1.57050684715428305011054075461, 3.16537886852318153350254269490, 3.91037351924812594223985792086, 4.98624811114539665539354289203, 5.30852168681852901648024062341, 6.29083246312614561869923438872, 7.23713402589484232943368699180, 7.80032570571314572026789764339, 8.501951421340583370945141673758

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