Properties

Label 2-372810-1.1-c1-0-11
Degree $2$
Conductor $372810$
Sign $1$
Analytic cond. $2976.90$
Root an. cond. $54.5610$
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 + 7-s − 8-s + 9-s + 10-s − 2·11-s − 12-s + 5·13-s − 14-s + 15-s + 16-s − 18-s − 20-s − 21-s + 2·22-s − 4·23-s + 24-s + 25-s − 5·26-s − 27-s + 28-s − 2·29-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 + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s + 1.38·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.223·20-s − 0.218·21-s + 0.426·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.980·26-s − 0.192·27-s + 0.188·28-s − 0.371·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(372810\)    =    \(2 \cdot 3 \cdot 5 \cdot 17^{2} \cdot 43\)
Sign: $1$
Analytic conductor: \(2976.90\)
Root analytic conductor: \(54.5610\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 372810,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8289711147\)
\(L(\frac12)\) \(\approx\) \(0.8289711147\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
17 \( 1 \)
43 \( 1 - T \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−12.21808625873760, −12.00545452812050, −11.55088002973157, −10.95386648969001, −10.75537435427900, −10.43037181794465, −9.795391003866884, −9.327690822565850, −8.781010071497237, −8.382550532270109, −7.837428362119920, −7.708256545296235, −6.945168828262114, −6.526125809676472, −6.069378382043058, −5.562835916792654, −5.121378409687610, −4.425995460116119, −4.031209837024287, −3.322264015941654, −2.981955114910716, −1.983964534592848, −1.691075129422914, −0.9759171815616300, −0.3165686332861723, 0.3165686332861723, 0.9759171815616300, 1.691075129422914, 1.983964534592848, 2.981955114910716, 3.322264015941654, 4.031209837024287, 4.425995460116119, 5.121378409687610, 5.562835916792654, 6.069378382043058, 6.526125809676472, 6.945168828262114, 7.708256545296235, 7.837428362119920, 8.382550532270109, 8.781010071497237, 9.327690822565850, 9.795391003866884, 10.43037181794465, 10.75537435427900, 10.95386648969001, 11.55088002973157, 12.00545452812050, 12.21808625873760

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