Properties

Label 2-85176-1.1-c1-0-47
Degree $2$
Conductor $85176$
Sign $-1$
Analytic cond. $680.133$
Root an. cond. $26.0793$
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  − 7-s + 4·11-s + 6·17-s + 4·19-s − 4·23-s − 5·25-s − 2·29-s − 8·31-s − 6·37-s − 6·41-s + 4·43-s − 6·47-s + 49-s − 6·53-s + 12·59-s + 8·61-s − 6·67-s + 12·71-s + 2·73-s − 4·77-s − 12·79-s + 10·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.377·7-s + 1.20·11-s + 1.45·17-s + 0.917·19-s − 0.834·23-s − 25-s − 0.371·29-s − 1.43·31-s − 0.986·37-s − 0.937·41-s + 0.609·43-s − 0.875·47-s + 1/7·49-s − 0.824·53-s + 1.56·59-s + 1.02·61-s − 0.733·67-s + 1.42·71-s + 0.234·73-s − 0.455·77-s − 1.35·79-s + 1.05·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85176\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(680.133\)
Root analytic conductor: \(26.0793\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 85176,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
13 \( 1 \)
good5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.22175274843049, −13.80705554718181, −13.14333246234420, −12.69526939901231, −12.01933837320703, −11.84101908136966, −11.36263879776135, −10.65441933971293, −10.05328462170599, −9.592038901278159, −9.417446692732672, −8.584173039750427, −8.179843677991732, −7.414019198337977, −7.193160912718780, −6.462425629582821, −5.896366147401007, −5.477611442422132, −4.909996376212974, −3.988770140675691, −3.541768312979298, −3.322139734234169, −2.197741200380653, −1.646755337221945, −0.9550826466107077, 0, 0.9550826466107077, 1.646755337221945, 2.197741200380653, 3.322139734234169, 3.541768312979298, 3.988770140675691, 4.909996376212974, 5.477611442422132, 5.896366147401007, 6.462425629582821, 7.193160912718780, 7.414019198337977, 8.179843677991732, 8.584173039750427, 9.417446692732672, 9.592038901278159, 10.05328462170599, 10.65441933971293, 11.36263879776135, 11.84101908136966, 12.01933837320703, 12.69526939901231, 13.14333246234420, 13.80705554718181, 14.22175274843049

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