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Properties

Label 2-93600-1.1-c1-0-118
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  + 4·7-s + 4·11-s − 13-s + 2·17-s + 4·19-s − 6·29-s − 4·31-s − 6·37-s − 10·41-s − 8·43-s + 12·47-s + 9·49-s − 10·53-s − 12·59-s + 6·61-s − 4·67-s + 4·71-s − 2·73-s + 16·77-s − 8·79-s − 12·83-s + 6·89-s − 4·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 1.51·7-s + 1.20·11-s − 0.277·13-s + 0.485·17-s + 0.917·19-s − 1.11·29-s − 0.718·31-s − 0.986·37-s − 1.56·41-s − 1.21·43-s + 1.75·47-s + 9/7·49-s − 1.37·53-s − 1.56·59-s + 0.768·61-s − 0.488·67-s + 0.474·71-s − 0.234·73-s + 1.82·77-s − 0.900·79-s − 1.31·83-s + 0.635·89-s − 0.419·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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: \(1\)
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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.26257943966332, −13.72844409998812, −13.16815172200559, −12.40440533949877, −12.04110523375653, −11.57973603355666, −11.28715475548645, −10.67336215064220, −10.14785448203089, −9.539225171562580, −9.043269608288281, −8.622684019321793, −8.009270581301400, −7.525529626541718, −7.096882533892381, −6.534323974910602, −5.674938289613359, −5.385971877899843, −4.772654734375834, −4.258633725692317, −3.550485964382910, −3.139145780267019, −2.035605450109087, −1.618045414856728, −1.139543520713614, 0, 1.139543520713614, 1.618045414856728, 2.035605450109087, 3.139145780267019, 3.550485964382910, 4.258633725692317, 4.772654734375834, 5.385971877899843, 5.674938289613359, 6.534323974910602, 7.096882533892381, 7.525529626541718, 8.009270581301400, 8.622684019321793, 9.043269608288281, 9.539225171562580, 10.14785448203089, 10.67336215064220, 11.28715475548645, 11.57973603355666, 12.04110523375653, 12.40440533949877, 13.16815172200559, 13.72844409998812, 14.26257943966332

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