Properties

Label 2-388416-1.1-c1-0-155
Degree $2$
Conductor $388416$
Sign $-1$
Analytic cond. $3101.51$
Root an. cond. $55.6912$
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 − 2·5-s − 7-s + 9-s + 6·13-s − 2·15-s − 21-s − 8·23-s − 25-s + 27-s − 6·29-s − 8·31-s + 2·35-s + 10·37-s + 6·39-s + 6·41-s + 12·43-s − 2·45-s + 49-s + 10·53-s − 8·59-s + 6·61-s − 63-s − 12·65-s + 12·67-s − 8·69-s + 6·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.66·13-s − 0.516·15-s − 0.218·21-s − 1.66·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.338·35-s + 1.64·37-s + 0.960·39-s + 0.937·41-s + 1.82·43-s − 0.298·45-s + 1/7·49-s + 1.37·53-s − 1.04·59-s + 0.768·61-s − 0.125·63-s − 1.48·65-s + 1.46·67-s − 0.963·69-s + 0.702·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(388416\)    =    \(2^{6} \cdot 3 \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(3101.51\)
Root analytic conductor: \(55.6912\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{388416} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 388416,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 + T \)
17 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.69711794153869, −12.26446351028612, −11.80630175766129, −11.18286740464201, −10.99906366529111, −10.57551884619499, −9.853384155511719, −9.388291398983255, −9.140276923761291, −8.546012877795872, −8.043167529980835, −7.736755597827611, −7.419105914945529, −6.707327507483955, −6.205962102295279, −5.742353480614580, −5.424700249090430, −4.360402198429327, −4.022806519683329, −3.775682097830587, −3.390465621723384, −2.472139360114874, −2.194741065963087, −1.355471226134286, −0.7626791973209301, 0, 0.7626791973209301, 1.355471226134286, 2.194741065963087, 2.472139360114874, 3.390465621723384, 3.775682097830587, 4.022806519683329, 4.360402198429327, 5.424700249090430, 5.742353480614580, 6.205962102295279, 6.707327507483955, 7.419105914945529, 7.736755597827611, 8.043167529980835, 8.546012877795872, 9.140276923761291, 9.388291398983255, 9.853384155511719, 10.57551884619499, 10.99906366529111, 11.18286740464201, 11.80630175766129, 12.26446351028612, 12.69711794153869

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