Properties

Label 2-3630-1.1-c1-0-3
Degree $2$
Conductor $3630$
Sign $1$
Analytic cond. $28.9856$
Root an. cond. $5.38383$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s − 3·7-s − 8-s + 9-s + 10-s − 12-s − 13-s + 3·14-s + 15-s + 16-s + 7·17-s − 18-s + 5·19-s − 20-s + 3·21-s − 8·23-s + 24-s + 25-s + 26-s − 27-s − 3·28-s + 5·29-s − 30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.277·13-s + 0.801·14-s + 0.258·15-s + 1/4·16-s + 1.69·17-s − 0.235·18-s + 1.14·19-s − 0.223·20-s + 0.654·21-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.566·28-s + 0.928·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3630\)    =    \(2 \cdot 3 \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(28.9856\)
Root analytic conductor: \(5.38383\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3630,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5974067793\)
\(L(\frac12)\) \(\approx\) \(0.5974067793\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 + T \)
11 \( 1 \)
good7 \( 1 + 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.434619325514851110490555148572, −7.77049789872902778638533918466, −7.17597688792461810600841461991, −6.36411715944922490710130608918, −5.73696462087675392990589139672, −4.89758905800619126505365147612, −3.58345532764719452575260203850, −3.17534707682927881184223136074, −1.73080658302757130293409278688, −0.51546057438666302322702554102, 0.51546057438666302322702554102, 1.73080658302757130293409278688, 3.17534707682927881184223136074, 3.58345532764719452575260203850, 4.89758905800619126505365147612, 5.73696462087675392990589139672, 6.36411715944922490710130608918, 7.17597688792461810600841461991, 7.77049789872902778638533918466, 8.434619325514851110490555148572

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