Properties

Label 2-93600-1.1-c1-0-71
Degree $2$
Conductor $93600$
Sign $-1$
Analytic cond. $747.399$
Root an. cond. $27.3386$
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 − 3·11-s + 13-s − 7·17-s + 8·19-s + 4·23-s + 3·29-s + 11·31-s + 2·41-s + 8·43-s − 9·47-s + 2·49-s − 9·53-s − 9·59-s + 61-s − 5·67-s − 12·73-s + 9·77-s + 8·79-s − 9·83-s − 12·89-s − 3·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 1.13·7-s − 0.904·11-s + 0.277·13-s − 1.69·17-s + 1.83·19-s + 0.834·23-s + 0.557·29-s + 1.97·31-s + 0.312·41-s + 1.21·43-s − 1.31·47-s + 2/7·49-s − 1.23·53-s − 1.17·59-s + 0.128·61-s − 0.610·67-s − 1.40·73-s + 1.02·77-s + 0.900·79-s − 0.987·83-s − 1.27·89-s − 0.314·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93600\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(747.399\)
Root analytic conductor: \(27.3386\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{93600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 93600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 11 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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

−13.97038575903942, −13.47370638772228, −13.12792140803411, −12.75267439282122, −12.15626841033897, −11.51215338994822, −11.22113797098853, −10.51804703771297, −10.13589478591845, −9.574676502985514, −9.157013862690961, −8.685213438469747, −7.949860735370696, −7.582657362492929, −6.861661363032583, −6.477022587511350, −6.017660191554568, −5.317923882955673, −4.701694479333378, −4.336421354913599, −3.365345732040736, −2.890184901881839, −2.644462510671464, −1.568899194019320, −0.7870039817200973, 0, 0.7870039817200973, 1.568899194019320, 2.644462510671464, 2.890184901881839, 3.365345732040736, 4.336421354913599, 4.701694479333378, 5.317923882955673, 6.017660191554568, 6.477022587511350, 6.861661363032583, 7.582657362492929, 7.949860735370696, 8.685213438469747, 9.157013862690961, 9.574676502985514, 10.13589478591845, 10.51804703771297, 11.22113797098853, 11.51215338994822, 12.15626841033897, 12.75267439282122, 13.12792140803411, 13.47370638772228, 13.97038575903942

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