Properties

Label 2-3630-1.1-c1-0-63
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 + 1.85·13-s − 3.61·14-s − 15-s + 16-s − 2.47·17-s + 18-s + 1.85·19-s + 20-s + 3.61·21-s − 6.38·23-s − 24-s + 25-s + 1.85·26-s − 27-s − 3.61·28-s + 2.76·29-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 + 0.514·13-s − 0.966·14-s − 0.258·15-s + 0.250·16-s − 0.599·17-s + 0.235·18-s + 0.425·19-s + 0.223·20-s + 0.789·21-s − 1.33·23-s − 0.204·24-s + 0.200·25-s + 0.363·26-s − 0.192·27-s − 0.683·28-s + 0.513·29-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 \( 1 - 1.85T + 13T^{2} \)
17 \( 1 + 2.47T + 17T^{2} \)
19 \( 1 - 1.85T + 19T^{2} \)
23 \( 1 + 6.38T + 23T^{2} \)
29 \( 1 - 2.76T + 29T^{2} \)
31 \( 1 + 2.76T + 31T^{2} \)
37 \( 1 + 9.85T + 37T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 + 3.23T + 43T^{2} \)
47 \( 1 + 12.7T + 47T^{2} \)
53 \( 1 + 1.90T + 53T^{2} \)
59 \( 1 - 6.32T + 59T^{2} \)
61 \( 1 - 9.70T + 61T^{2} \)
67 \( 1 + 2.29T + 67T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 - 2.47T + 83T^{2} \)
89 \( 1 + 8.56T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−8.099539965585419225973610808972, −7.00065817729785102810165062017, −6.57276791632986721839295974336, −5.90747957522273581367819695450, −5.33159486445693684658443115418, −4.28079696463934018877004978494, −3.55841137571843737698379049480, −2.68376304686716637507400671629, −1.55769799043235622931016301067, 0, 1.55769799043235622931016301067, 2.68376304686716637507400671629, 3.55841137571843737698379049480, 4.28079696463934018877004978494, 5.33159486445693684658443115418, 5.90747957522273581367819695450, 6.57276791632986721839295974336, 7.00065817729785102810165062017, 8.099539965585419225973610808972

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