Properties

Label 2-3675-1.1-c1-0-83
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  + 2.73·2-s − 3-s + 5.46·4-s − 2.73·6-s + 9.46·8-s + 9-s + 0.732·11-s − 5.46·12-s + 2.26·13-s + 14.9·16-s + 3.26·17-s + 2.73·18-s − 4.46·19-s + 2·22-s + 4.73·23-s − 9.46·24-s + 6.19·26-s − 27-s − 4.19·29-s + 0.464·31-s + 21.8·32-s − 0.732·33-s + 8.92·34-s + 5.46·36-s + 3.19·37-s − 12.1·38-s − 2.26·39-s + ⋯
L(s)  = 1  + 1.93·2-s − 0.577·3-s + 2.73·4-s − 1.11·6-s + 3.34·8-s + 0.333·9-s + 0.220·11-s − 1.57·12-s + 0.629·13-s + 3.73·16-s + 0.792·17-s + 0.643·18-s − 1.02·19-s + 0.426·22-s + 0.986·23-s − 1.93·24-s + 1.21·26-s − 0.192·27-s − 0.779·29-s + 0.0833·31-s + 3.86·32-s − 0.127·33-s + 1.53·34-s + 0.910·36-s + 0.525·37-s − 1.97·38-s − 0.363·39-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

\(L(1)\) \(\approx\) \(6.312685646\)
\(L(\frac12)\) \(\approx\) \(6.312685646\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.73T + 2T^{2} \)
11 \( 1 - 0.732T + 11T^{2} \)
13 \( 1 - 2.26T + 13T^{2} \)
17 \( 1 - 3.26T + 17T^{2} \)
19 \( 1 + 4.46T + 19T^{2} \)
23 \( 1 - 4.73T + 23T^{2} \)
29 \( 1 + 4.19T + 29T^{2} \)
31 \( 1 - 0.464T + 31T^{2} \)
37 \( 1 - 3.19T + 37T^{2} \)
41 \( 1 - 0.732T + 41T^{2} \)
43 \( 1 + 3.19T + 43T^{2} \)
47 \( 1 - 2T + 47T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 - 0.196T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 - 14.6T + 67T^{2} \)
71 \( 1 - 6.19T + 71T^{2} \)
73 \( 1 - 12.6T + 73T^{2} \)
79 \( 1 + 7.39T + 79T^{2} \)
83 \( 1 - 15.1T + 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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

−8.190415223690661105187529285367, −7.43940302657860016755960654737, −6.57573638910107396292674728378, −6.21887629193999175199005858561, −5.37673462065514077764130003734, −4.82842337953180302293122281503, −3.95251876895028060613112539802, −3.36415960026344400067495415761, −2.30177497798395261531748571307, −1.26088179737818998207435618996, 1.26088179737818998207435618996, 2.30177497798395261531748571307, 3.36415960026344400067495415761, 3.95251876895028060613112539802, 4.82842337953180302293122281503, 5.37673462065514077764130003734, 6.21887629193999175199005858561, 6.57573638910107396292674728378, 7.43940302657860016755960654737, 8.190415223690661105187529285367

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