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Properties

Label 2-193200-1.1-c1-0-113
Degree $2$
Conductor $193200$
Sign $-1$
Analytic cond. $1542.70$
Root an. cond. $39.2773$
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  + 3-s + 7-s + 9-s − 2·13-s − 6·17-s − 4·19-s + 21-s − 23-s + 27-s + 6·29-s − 4·31-s − 2·37-s − 2·39-s + 10·41-s − 8·43-s + 4·47-s + 49-s − 6·51-s + 6·53-s − 4·57-s + 4·59-s + 2·61-s + 63-s − 8·67-s − 69-s + 8·71-s − 10·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.218·21-s − 0.208·23-s + 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.328·37-s − 0.320·39-s + 1.56·41-s − 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.840·51-s + 0.824·53-s − 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.125·63-s − 0.977·67-s − 0.120·69-s + 0.949·71-s − 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(193200\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(1542.70\)
Root analytic conductor: \(39.2773\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 193200,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 - T \)
23 \( 1 + T \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−13.36444127963215, −12.87221004553828, −12.51220297461146, −11.90396000462769, −11.48810068397999, −10.91867882538481, −10.47416528942661, −10.12043497377654, −9.445073532786436, −8.943282812743293, −8.663193538238817, −8.164926834223324, −7.606610184535410, −7.107786402307675, −6.659699830575174, −6.145500879961099, −5.522765960818216, −4.803488793626598, −4.490411714837564, −3.978530636546421, −3.362655657928534, −2.543592121594484, −2.277353325608136, −1.695871263133605, −0.8130098558543632, 0, 0.8130098558543632, 1.695871263133605, 2.277353325608136, 2.543592121594484, 3.362655657928534, 3.978530636546421, 4.490411714837564, 4.803488793626598, 5.522765960818216, 6.145500879961099, 6.659699830575174, 7.107786402307675, 7.606610184535410, 8.164926834223324, 8.663193538238817, 8.943282812743293, 9.445073532786436, 10.12043497377654, 10.47416528942661, 10.91867882538481, 11.48810068397999, 11.90396000462769, 12.51220297461146, 12.87221004553828, 13.36444127963215

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