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Properties

Label 2-88200-1.1-c1-0-179
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  + 6·13-s + 2·17-s − 4·19-s + 4·23-s − 6·29-s − 6·37-s − 2·41-s + 4·43-s − 8·47-s − 2·53-s − 12·59-s − 6·61-s + 4·67-s + 12·71-s + 10·73-s − 8·79-s + 12·83-s + 14·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.66·13-s + 0.485·17-s − 0.917·19-s + 0.834·23-s − 1.11·29-s − 0.986·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s − 0.274·53-s − 1.56·59-s − 0.768·61-s + 0.488·67-s + 1.42·71-s + 1.17·73-s − 0.900·79-s + 1.31·83-s + 1.48·89-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 \( 1 + p T^{2} \) 1.11.a
13 \( 1 - 6 T + p T^{2} \) 1.13.ag
17 \( 1 - 2 T + p T^{2} \) 1.17.ac
19 \( 1 + 4 T + p T^{2} \) 1.19.e
23 \( 1 - 4 T + p T^{2} \) 1.23.ae
29 \( 1 + 6 T + p T^{2} \) 1.29.g
31 \( 1 + p T^{2} \) 1.31.a
37 \( 1 + 6 T + p T^{2} \) 1.37.g
41 \( 1 + 2 T + p T^{2} \) 1.41.c
43 \( 1 - 4 T + p T^{2} \) 1.43.ae
47 \( 1 + 8 T + p T^{2} \) 1.47.i
53 \( 1 + 2 T + p T^{2} \) 1.53.c
59 \( 1 + 12 T + p T^{2} \) 1.59.m
61 \( 1 + 6 T + p T^{2} \) 1.61.g
67 \( 1 - 4 T + p T^{2} \) 1.67.ae
71 \( 1 - 12 T + p T^{2} \) 1.71.am
73 \( 1 - 10 T + p T^{2} \) 1.73.ak
79 \( 1 + 8 T + p T^{2} \) 1.79.i
83 \( 1 - 12 T + p T^{2} \) 1.83.am
89 \( 1 - 14 T + p T^{2} \) 1.89.ao
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

−13.98368151167920, −13.71418819641157, −13.08756877411854, −12.74653077764400, −12.25744394403634, −11.55352000247456, −11.10752984901368, −10.73036480240093, −10.33570535772501, −9.503815911672814, −9.138191390659689, −8.646603607334598, −8.093766563030481, −7.670749984139113, −6.949703947407890, −6.354405464739457, −6.100550771946813, −5.308933154431321, −4.890897162254159, −4.094007516441233, −3.539620676261821, −3.203934295942957, −2.227813590531845, −1.613855416925463, −0.9742294260013086, 0, 0.9742294260013086, 1.613855416925463, 2.227813590531845, 3.203934295942957, 3.539620676261821, 4.094007516441233, 4.890897162254159, 5.308933154431321, 6.100550771946813, 6.354405464739457, 6.949703947407890, 7.670749984139113, 8.093766563030481, 8.646603607334598, 9.138191390659689, 9.503815911672814, 10.33570535772501, 10.73036480240093, 11.10752984901368, 11.55352000247456, 12.25744394403634, 12.74653077764400, 13.08756877411854, 13.71418819641157, 13.98368151167920

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