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Properties

Label 2-93600-1.1-c1-0-125
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 4·11-s − 13-s − 6·19-s − 6·23-s + 8·29-s − 10·37-s + 10·41-s + 12·43-s − 12·47-s − 3·49-s − 6·53-s − 12·59-s − 10·61-s + 4·67-s + 8·71-s + 4·73-s + 8·77-s − 8·79-s − 4·83-s − 10·89-s + 2·91-s + 4·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.755·7-s − 1.20·11-s − 0.277·13-s − 1.37·19-s − 1.25·23-s + 1.48·29-s − 1.64·37-s + 1.56·41-s + 1.82·43-s − 1.75·47-s − 3/7·49-s − 0.824·53-s − 1.56·59-s − 1.28·61-s + 0.488·67-s + 0.949·71-s + 0.468·73-s + 0.911·77-s − 0.900·79-s − 0.439·83-s − 1.05·89-s + 0.209·91-s + 0.406·97-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−14.12447532426157, −13.95413519976161, −13.28638797167768, −12.65904927613952, −12.48292252998155, −12.14652178153399, −11.12864693624875, −10.89679948402773, −10.33716669443969, −9.907016684629342, −9.460425285242240, −8.791181566808500, −8.246944867946711, −7.857766285492980, −7.316228561415700, −6.536519206728834, −6.268206534916724, −5.714821264536531, −4.978537811531884, −4.509706137935472, −3.928984679667739, −3.161009834941977, −2.654947248582163, −2.134861331601748, −1.285584268756011, 0, 0, 1.285584268756011, 2.134861331601748, 2.654947248582163, 3.161009834941977, 3.928984679667739, 4.509706137935472, 4.978537811531884, 5.714821264536531, 6.268206534916724, 6.536519206728834, 7.316228561415700, 7.857766285492980, 8.246944867946711, 8.791181566808500, 9.460425285242240, 9.907016684629342, 10.33716669443969, 10.89679948402773, 11.12864693624875, 12.14652178153399, 12.48292252998155, 12.65904927613952, 13.28638797167768, 13.95413519976161, 14.12447532426157

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