Properties

Label 2-3630-1.1-c1-0-59
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 12-s − 2·13-s + 15-s + 16-s + 2·17-s + 18-s − 8·19-s − 20-s + 4·23-s − 24-s + 25-s − 2·26-s − 27-s − 2·29-s + 30-s + 8·31-s + 32-s + 2·34-s + 36-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.554·13-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 1.83·19-s − 0.223·20-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.371·29-s + 0.182·30-s + 1.43·31-s + 0.176·32-s + 0.342·34-s + 1/6·36-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: \(1\)
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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−8.133047174223284885537321962931, −7.16836230024370529685281879263, −6.64185844480649323327260602992, −5.91684194713421278908131990725, −4.96078481994846767017064839056, −4.52372061623460830421897906486, −3.58856932177347037443591355642, −2.66852833357594314054225540316, −1.51848524228770769025148709326, 0, 1.51848524228770769025148709326, 2.66852833357594314054225540316, 3.58856932177347037443591355642, 4.52372061623460830421897906486, 4.96078481994846767017064839056, 5.91684194713421278908131990725, 6.64185844480649323327260602992, 7.16836230024370529685281879263, 8.133047174223284885537321962931

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