Properties

Label 2-93600-1.1-c1-0-86
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 + 5·17-s − 6·19-s + 3·23-s + 2·29-s + 6·31-s − 3·37-s − 5·41-s + 2·49-s − 7·53-s + 61-s + 15·71-s + 2·73-s − 9·77-s − 15·79-s + 6·83-s − 11·89-s − 3·91-s − 97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 15·119-s + ⋯
L(s)  = 1  − 1.13·7-s + 0.904·11-s + 0.277·13-s + 1.21·17-s − 1.37·19-s + 0.625·23-s + 0.371·29-s + 1.07·31-s − 0.493·37-s − 0.780·41-s + 2/7·49-s − 0.961·53-s + 0.128·61-s + 1.78·71-s + 0.234·73-s − 1.02·77-s − 1.68·79-s + 0.658·83-s − 1.16·89-s − 0.314·91-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 1.37·119-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 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 7 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−13.98336141292075, −13.65597254660205, −12.96710151418259, −12.59974947715561, −12.23747158816212, −11.66352950737982, −11.13739677158215, −10.51889928073487, −10.07404011496276, −9.637154731431260, −9.172999664754719, −8.503631753746668, −8.236346736400047, −7.458679917713898, −6.705194455096142, −6.593354957698913, −6.028070165262989, −5.409220833818132, −4.704627599219264, −4.122128525290746, −3.482619221325362, −3.126822515685628, −2.390951521476763, −1.539915678157374, −0.8933183927289925, 0, 0.8933183927289925, 1.539915678157374, 2.390951521476763, 3.126822515685628, 3.482619221325362, 4.122128525290746, 4.704627599219264, 5.409220833818132, 6.028070165262989, 6.593354957698913, 6.705194455096142, 7.458679917713898, 8.236346736400047, 8.503631753746668, 9.172999664754719, 9.637154731431260, 10.07404011496276, 10.51889928073487, 11.13739677158215, 11.66352950737982, 12.23747158816212, 12.59974947715561, 12.96710151418259, 13.65597254660205, 13.98336141292075

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