Properties

Label 2-29400-1.1-c1-0-21
Degree $2$
Conductor $29400$
Sign $1$
Analytic cond. $234.760$
Root an. cond. $15.3218$
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  + 3-s + 9-s − 13-s − 17-s − 6·19-s − 9·23-s + 27-s + 7·29-s + 7·31-s − 4·37-s − 39-s − 7·41-s + 43-s + 12·47-s − 51-s + 9·53-s − 6·57-s + 3·59-s − 9·61-s + 4·67-s − 9·69-s − 12·71-s + 8·73-s + 4·79-s + 81-s − 9·83-s + 7·87-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.277·13-s − 0.242·17-s − 1.37·19-s − 1.87·23-s + 0.192·27-s + 1.29·29-s + 1.25·31-s − 0.657·37-s − 0.160·39-s − 1.09·41-s + 0.152·43-s + 1.75·47-s − 0.140·51-s + 1.23·53-s − 0.794·57-s + 0.390·59-s − 1.15·61-s + 0.488·67-s − 1.08·69-s − 1.42·71-s + 0.936·73-s + 0.450·79-s + 1/9·81-s − 0.987·83-s + 0.750·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29400\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(234.760\)
Root analytic conductor: \(15.3218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 29400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.157440224\)
\(L(\frac12)\) \(\approx\) \(2.157440224\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 9 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−15.27580501037923, −14.58428085281860, −14.01254436912012, −13.71836170432789, −13.16801859613761, −12.32305518907114, −12.15887199679326, −11.55960991876648, −10.60926089100795, −10.28365923109415, −9.924200890388075, −9.006231858160999, −8.628689455542750, −8.125112758045579, −7.582765013067211, −6.779858934817419, −6.378214431485824, −5.713326090873763, −4.864330671867758, −4.233974995996132, −3.842590971002202, −2.843535666556783, −2.330822687715843, −1.649881199524883, −0.5297171112072957, 0.5297171112072957, 1.649881199524883, 2.330822687715843, 2.843535666556783, 3.842590971002202, 4.233974995996132, 4.864330671867758, 5.713326090873763, 6.378214431485824, 6.779858934817419, 7.582765013067211, 8.125112758045579, 8.628689455542750, 9.006231858160999, 9.924200890388075, 10.28365923109415, 10.60926089100795, 11.55960991876648, 12.15887199679326, 12.32305518907114, 13.16801859613761, 13.71836170432789, 14.01254436912012, 14.58428085281860, 15.27580501037923

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