Properties

Label 2-3675-1.1-c1-0-33
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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 1.61·2-s − 3-s + 0.618·4-s − 1.61·6-s − 2.23·8-s + 9-s − 2.23·11-s − 0.618·12-s + 6.47·13-s − 4.85·16-s − 2.47·17-s + 1.61·18-s − 2.47·19-s − 3.61·22-s + 4.23·23-s + 2.23·24-s + 10.4·26-s − 27-s − 3·29-s + 4·31-s − 3.38·32-s + 2.23·33-s − 4.00·34-s + 0.618·36-s − 3.47·37-s − 4.00·38-s − 6.47·39-s + ⋯
L(s)  = 1  + 1.14·2-s − 0.577·3-s + 0.309·4-s − 0.660·6-s − 0.790·8-s + 0.333·9-s − 0.674·11-s − 0.178·12-s + 1.79·13-s − 1.21·16-s − 0.599·17-s + 0.381·18-s − 0.567·19-s − 0.771·22-s + 0.883·23-s + 0.456·24-s + 2.05·26-s − 0.192·27-s − 0.557·29-s + 0.718·31-s − 0.597·32-s + 0.389·33-s − 0.685·34-s + 0.103·36-s − 0.570·37-s − 0.648·38-s − 1.03·39-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

\(L(1)\) \(\approx\) \(2.371935906\)
\(L(\frac12)\) \(\approx\) \(2.371935906\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.61T + 2T^{2} \)
11 \( 1 + 2.23T + 11T^{2} \)
13 \( 1 - 6.47T + 13T^{2} \)
17 \( 1 + 2.47T + 17T^{2} \)
19 \( 1 + 2.47T + 19T^{2} \)
23 \( 1 - 4.23T + 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 3.47T + 37T^{2} \)
41 \( 1 - 8.94T + 41T^{2} \)
43 \( 1 - 4.70T + 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 6.47T + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 + 12.7T + 67T^{2} \)
71 \( 1 - 8.23T + 71T^{2} \)
73 \( 1 + 14.4T + 73T^{2} \)
79 \( 1 + 2.70T + 79T^{2} \)
83 \( 1 - 16.9T + 83T^{2} \)
89 \( 1 - 1.52T + 89T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.716424920937127424608902795774, −7.62825275732571359821152060969, −6.70647755003476607030460074262, −6.03403930986436437592101397190, −5.58518575015332333207613494251, −4.67505048655027284685722919623, −4.07331552786894921182291248463, −3.27086315745740890875196620470, −2.24724952467315039669351320518, −0.77847757740791034949696327729, 0.77847757740791034949696327729, 2.24724952467315039669351320518, 3.27086315745740890875196620470, 4.07331552786894921182291248463, 4.67505048655027284685722919623, 5.58518575015332333207613494251, 6.03403930986436437592101397190, 6.70647755003476607030460074262, 7.62825275732571359821152060969, 8.716424920937127424608902795774

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