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Properties

Label 2-88200-1.1-c1-0-176
Degree $2$
Conductor $88200$
Sign $-1$
Analytic cond. $704.280$
Root an. cond. $26.5382$
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  + 3·11-s + 2·13-s − 2·19-s − 7·23-s + 3·29-s + 6·31-s − 3·37-s + 5·43-s + 2·47-s − 2·53-s − 10·59-s + 8·61-s − 9·67-s − 9·71-s + 8·73-s − 79-s + 14·83-s − 6·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.904·11-s + 0.554·13-s − 0.458·19-s − 1.45·23-s + 0.557·29-s + 1.07·31-s − 0.493·37-s + 0.762·43-s + 0.291·47-s − 0.274·53-s − 1.30·59-s + 1.02·61-s − 1.09·67-s − 1.06·71-s + 0.936·73-s − 0.112·79-s + 1.53·83-s − 0.635·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88200\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(704.280\)
Root analytic conductor: \(26.5382\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 88200,\ (\ :1/2),\ -1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 - 3 T + p T^{2} \) 1.11.ad
13 \( 1 - 2 T + p T^{2} \) 1.13.ac
17 \( 1 + p T^{2} \) 1.17.a
19 \( 1 + 2 T + p T^{2} \) 1.19.c
23 \( 1 + 7 T + p T^{2} \) 1.23.h
29 \( 1 - 3 T + p T^{2} \) 1.29.ad
31 \( 1 - 6 T + p T^{2} \) 1.31.ag
37 \( 1 + 3 T + p T^{2} \) 1.37.d
41 \( 1 + p T^{2} \) 1.41.a
43 \( 1 - 5 T + p T^{2} \) 1.43.af
47 \( 1 - 2 T + p T^{2} \) 1.47.ac
53 \( 1 + 2 T + p T^{2} \) 1.53.c
59 \( 1 + 10 T + p T^{2} \) 1.59.k
61 \( 1 - 8 T + p T^{2} \) 1.61.ai
67 \( 1 + 9 T + p T^{2} \) 1.67.j
71 \( 1 + 9 T + p T^{2} \) 1.71.j
73 \( 1 - 8 T + p T^{2} \) 1.73.ai
79 \( 1 + T + p T^{2} \) 1.79.b
83 \( 1 - 14 T + p T^{2} \) 1.83.ao
89 \( 1 + 6 T + p T^{2} \) 1.89.g
97 \( 1 - 2 T + p T^{2} \) 1.97.ac
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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

−14.07515802582875, −13.71399232121773, −13.27886146936721, −12.46132326370849, −12.21735664153562, −11.72752032825460, −11.22646410846464, −10.58575819214218, −10.23632461507624, −9.658298996208644, −9.041738433895392, −8.728158727140486, −7.974547875935955, −7.779371532782981, −6.827809814068368, −6.492214958032282, −6.019777611411776, −5.465040686617532, −4.641329652541026, −4.183821124882636, −3.723652224828341, −2.992825310213674, −2.309115163490080, −1.595162336885481, −0.9727315260328322, 0, 0.9727315260328322, 1.595162336885481, 2.309115163490080, 2.992825310213674, 3.723652224828341, 4.183821124882636, 4.641329652541026, 5.465040686617532, 6.019777611411776, 6.492214958032282, 6.827809814068368, 7.779371532782981, 7.974547875935955, 8.728158727140486, 9.041738433895392, 9.658298996208644, 10.23632461507624, 10.58575819214218, 11.22646410846464, 11.72752032825460, 12.21735664153562, 12.46132326370849, 13.27886146936721, 13.71399232121773, 14.07515802582875

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