Properties

Label 2-1638-1.1-c3-0-27
Degree $2$
Conductor $1638$
Sign $1$
Analytic cond. $96.6451$
Root an. cond. $9.83082$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·4-s + 7·7-s + 8·8-s − 39·11-s + 13·13-s + 14·14-s + 16·16-s − 24·17-s + 38·19-s − 78·22-s − 39·23-s − 125·25-s + 26·26-s + 28·28-s + 96·29-s + 227·31-s + 32·32-s − 48·34-s + 425·37-s + 76·38-s + 105·41-s + 344·43-s − 156·44-s − 78·46-s − 99·47-s + 49·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1.06·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.342·17-s + 0.458·19-s − 0.755·22-s − 0.353·23-s − 25-s + 0.196·26-s + 0.188·28-s + 0.614·29-s + 1.31·31-s + 0.176·32-s − 0.242·34-s + 1.88·37-s + 0.324·38-s + 0.399·41-s + 1.21·43-s − 0.534·44-s − 0.250·46-s − 0.307·47-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(96.6451\)
Root analytic conductor: \(9.83082\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1638} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.531806527\)
\(L(\frac12)\) \(\approx\) \(3.531806527\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 \)
7 \( 1 - p T \)
13 \( 1 - p T \)
good5 \( 1 + p^{3} T^{2} \)
11 \( 1 + 39 T + p^{3} T^{2} \)
17 \( 1 + 24 T + p^{3} T^{2} \)
19 \( 1 - 2 p T + p^{3} T^{2} \)
23 \( 1 + 39 T + p^{3} T^{2} \)
29 \( 1 - 96 T + p^{3} T^{2} \)
31 \( 1 - 227 T + p^{3} T^{2} \)
37 \( 1 - 425 T + p^{3} T^{2} \)
41 \( 1 - 105 T + p^{3} T^{2} \)
43 \( 1 - 8 p T + p^{3} T^{2} \)
47 \( 1 + 99 T + p^{3} T^{2} \)
53 \( 1 - 540 T + p^{3} T^{2} \)
59 \( 1 + 114 T + p^{3} T^{2} \)
61 \( 1 + 565 T + p^{3} T^{2} \)
67 \( 1 + 385 T + p^{3} T^{2} \)
71 \( 1 - 156 T + p^{3} T^{2} \)
73 \( 1 + 673 T + p^{3} T^{2} \)
79 \( 1 - 749 T + p^{3} T^{2} \)
83 \( 1 - 1044 T + p^{3} T^{2} \)
89 \( 1 - 690 T + p^{3} T^{2} \)
97 \( 1 - 317 T + p^{3} T^{2} \)
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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

−8.980021207538217101705670370264, −7.911220580499200874852251006618, −7.60188821443171453953604110924, −6.35140665735117345858229721516, −5.76386307522229898423730883221, −4.80343392632358320918035194111, −4.14214424718886237920003011473, −2.95174984679220826217720905004, −2.17880442447841531528081046751, −0.809051335872597004219867188697, 0.809051335872597004219867188697, 2.17880442447841531528081046751, 2.95174984679220826217720905004, 4.14214424718886237920003011473, 4.80343392632358320918035194111, 5.76386307522229898423730883221, 6.35140665735117345858229721516, 7.60188821443171453953604110924, 7.911220580499200874852251006618, 8.980021207538217101705670370264

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