Properties

Label 2-18354-1.1-c1-0-1
Degree $2$
Conductor $18354$
Sign $1$
Analytic cond. $146.557$
Root an. cond. $12.1060$
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 − 6-s − 7-s + 8-s + 9-s + 4·11-s − 12-s − 3·13-s − 14-s + 16-s − 5·17-s + 18-s − 19-s + 21-s + 4·22-s − 23-s − 24-s − 5·25-s − 3·26-s − 27-s − 28-s − 10·29-s + 4·31-s + 32-s − 4·33-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 0.832·13-s − 0.267·14-s + 1/4·16-s − 1.21·17-s + 0.235·18-s − 0.229·19-s + 0.218·21-s + 0.852·22-s − 0.208·23-s − 0.204·24-s − 25-s − 0.588·26-s − 0.192·27-s − 0.188·28-s − 1.85·29-s + 0.718·31-s + 0.176·32-s − 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18354\)    =    \(2 \cdot 3 \cdot 7 \cdot 19 \cdot 23\)
Sign: $1$
Analytic conductor: \(146.557\)
Root analytic conductor: \(12.1060\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 18354,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.999037218\)
\(L(\frac12)\) \(\approx\) \(1.999037218\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 + T \)
good5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−15.58913611673910, −15.33062882172518, −14.68961751433458, −14.07486132697441, −13.56549202738657, −13.02983074020449, −12.40173013577355, −11.94781456851894, −11.50237378890453, −10.92752531554042, −10.30236662548621, −9.602442186642584, −9.172810754237444, −8.412161967889895, −7.513505793987145, −6.953421390086885, −6.556012218232041, −5.839730935088926, −5.343863002901317, −4.489419770783477, −4.008428035164818, −3.427072022607755, −2.272404371254269, −1.817685042846408, −0.5257746861502579, 0.5257746861502579, 1.817685042846408, 2.272404371254269, 3.427072022607755, 4.008428035164818, 4.489419770783477, 5.343863002901317, 5.839730935088926, 6.556012218232041, 6.953421390086885, 7.513505793987145, 8.412161967889895, 9.172810754237444, 9.602442186642584, 10.30236662548621, 10.92752531554042, 11.50237378890453, 11.94781456851894, 12.40173013577355, 13.02983074020449, 13.56549202738657, 14.07486132697441, 14.68961751433458, 15.33062882172518, 15.58913611673910

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