Properties

Label 2-3630-1.1-c1-0-53
Degree $2$
Conductor $3630$
Sign $-1$
Analytic cond. $28.9856$
Root an. cond. $5.38383$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s − 3.61·7-s + 8-s + 9-s − 10-s − 12-s − 0.618·13-s − 3.61·14-s + 15-s + 16-s + 6.47·17-s + 18-s + 0.618·19-s − 20-s + 3.61·21-s + 4.85·23-s − 24-s + 25-s − 0.618·26-s − 27-s − 3.61·28-s − 3.70·29-s + 30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 1.36·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.288·12-s − 0.171·13-s − 0.966·14-s + 0.258·15-s + 0.250·16-s + 1.56·17-s + 0.235·18-s + 0.141·19-s − 0.223·20-s + 0.789·21-s + 1.01·23-s − 0.204·24-s + 0.200·25-s − 0.121·26-s − 0.192·27-s − 0.683·28-s − 0.688·29-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3630,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.61T + 7T^{2} \)
13 \( 1 + 0.618T + 13T^{2} \)
17 \( 1 - 6.47T + 17T^{2} \)
19 \( 1 - 0.618T + 19T^{2} \)
23 \( 1 - 4.85T + 23T^{2} \)
29 \( 1 + 3.70T + 29T^{2} \)
31 \( 1 + 9.23T + 31T^{2} \)
37 \( 1 + 0.618T + 37T^{2} \)
41 \( 1 - 9.32T + 41T^{2} \)
43 \( 1 + 0.763T + 43T^{2} \)
47 \( 1 + 2.61T + 47T^{2} \)
53 \( 1 + 6.09T + 53T^{2} \)
59 \( 1 + 14.6T + 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 + 8.18T + 67T^{2} \)
71 \( 1 + 0.291T + 71T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 - 4.47T + 79T^{2} \)
83 \( 1 - 4.94T + 83T^{2} \)
89 \( 1 + 5.14T + 89T^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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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

−7.75990508175538024717433768943, −7.38031093531554931148070203961, −6.54461564772389815547244424725, −5.85543514909425100242439875061, −5.26944688824655288825594300211, −4.29945039865810197247479018627, −3.41365641417666246485100237491, −2.95154696503534740493843135233, −1.40015458791826212768109164545, 0, 1.40015458791826212768109164545, 2.95154696503534740493843135233, 3.41365641417666246485100237491, 4.29945039865810197247479018627, 5.26944688824655288825594300211, 5.85543514909425100242439875061, 6.54461564772389815547244424725, 7.38031093531554931148070203961, 7.75990508175538024717433768943

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