Properties

Label 2-360e2-1.1-c1-0-137
Degree $2$
Conductor $129600$
Sign $-1$
Analytic cond. $1034.86$
Root an. cond. $32.1692$
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  − 3·7-s − 4·11-s + 3·17-s + 8·19-s + 2·29-s − 4·31-s + 8·37-s − 9·41-s − 6·43-s + 7·47-s + 2·49-s + 6·53-s − 12·59-s + 10·61-s + 10·67-s + 4·71-s − 9·73-s + 12·77-s + 5·79-s − 6·83-s + 13·89-s − 7·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.13·7-s − 1.20·11-s + 0.727·17-s + 1.83·19-s + 0.371·29-s − 0.718·31-s + 1.31·37-s − 1.40·41-s − 0.914·43-s + 1.02·47-s + 2/7·49-s + 0.824·53-s − 1.56·59-s + 1.28·61-s + 1.22·67-s + 0.474·71-s − 1.05·73-s + 1.36·77-s + 0.562·79-s − 0.658·83-s + 1.37·89-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129600\)    =    \(2^{6} \cdot 3^{4} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(1034.86\)
Root analytic conductor: \(32.1692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 129600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good7 \( 1 + 3 T + p T^{2} \) 1.7.d
11 \( 1 + 4 T + p T^{2} \) 1.11.e
13 \( 1 + p T^{2} \) 1.13.a
17 \( 1 - 3 T + p T^{2} \) 1.17.ad
19 \( 1 - 8 T + p T^{2} \) 1.19.ai
23 \( 1 + p T^{2} \) 1.23.a
29 \( 1 - 2 T + p T^{2} \) 1.29.ac
31 \( 1 + 4 T + p T^{2} \) 1.31.e
37 \( 1 - 8 T + p T^{2} \) 1.37.ai
41 \( 1 + 9 T + p T^{2} \) 1.41.j
43 \( 1 + 6 T + p T^{2} \) 1.43.g
47 \( 1 - 7 T + p T^{2} \) 1.47.ah
53 \( 1 - 6 T + p T^{2} \) 1.53.ag
59 \( 1 + 12 T + p T^{2} \) 1.59.m
61 \( 1 - 10 T + p T^{2} \) 1.61.ak
67 \( 1 - 10 T + p T^{2} \) 1.67.ak
71 \( 1 - 4 T + p T^{2} \) 1.71.ae
73 \( 1 + 9 T + p T^{2} \) 1.73.j
79 \( 1 - 5 T + p T^{2} \) 1.79.af
83 \( 1 + 6 T + p T^{2} \) 1.83.g
89 \( 1 - 13 T + p T^{2} \) 1.89.an
97 \( 1 + 7 T + p T^{2} \) 1.97.h
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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

−13.64762988335868, −13.22995919987784, −12.86999035879428, −12.25097415656973, −11.93943177208836, −11.33294226293665, −10.80341022370489, −10.17116355458595, −9.858122968684781, −9.569581087458207, −8.877888831525185, −8.343086754735106, −7.622812330905188, −7.503398878631471, −6.796917271446398, −6.268732938654333, −5.654842081444044, −5.226031029116703, −4.839687033536509, −3.792423516918658, −3.502864504796305, −2.830941195638654, −2.503399005547663, −1.482821136251700, −0.7737478840733393, 0, 0.7737478840733393, 1.482821136251700, 2.503399005547663, 2.830941195638654, 3.502864504796305, 3.792423516918658, 4.839687033536509, 5.226031029116703, 5.654842081444044, 6.268732938654333, 6.796917271446398, 7.503398878631471, 7.622812330905188, 8.343086754735106, 8.877888831525185, 9.569581087458207, 9.858122968684781, 10.17116355458595, 10.80341022370489, 11.33294226293665, 11.93943177208836, 12.25097415656973, 12.86999035879428, 13.22995919987784, 13.64762988335868

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