Properties

Label 2-93600-1.1-c1-0-13
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s − 5·11-s + 13-s + 17-s − 7·19-s + 4·23-s − 2·31-s − 8·37-s − 7·41-s + 12·43-s + 9·49-s − 10·53-s − 4·59-s + 8·61-s − 11·67-s − 12·71-s + 7·73-s − 20·77-s − 9·83-s − 9·89-s + 4·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.51·7-s − 1.50·11-s + 0.277·13-s + 0.242·17-s − 1.60·19-s + 0.834·23-s − 0.359·31-s − 1.31·37-s − 1.09·41-s + 1.82·43-s + 9/7·49-s − 1.37·53-s − 0.520·59-s + 1.02·61-s − 1.34·67-s − 1.42·71-s + 0.819·73-s − 2.27·77-s − 0.987·83-s − 0.953·89-s + 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

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

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: \(0\)
Selberg data: \((2,\ 93600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.738084583\)
\(L(\frac12)\) \(\approx\) \(1.738084583\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.95689580900155, −13.30317404155437, −12.77173676356873, −12.51752536412128, −11.80221321513061, −11.20405258338808, −10.87969114182495, −10.49521140551876, −10.09144001835740, −9.184580062174399, −8.681702274338670, −8.347173129953840, −7.814698207302899, −7.394747829579933, −6.813365062908112, −6.060237444281561, −5.485467963973738, −5.065816854305508, −4.550237596025976, −4.059708610788531, −3.167823504463245, −2.604397960692571, −1.892968878324331, −1.460729359892427, −0.4042235162244291, 0.4042235162244291, 1.460729359892427, 1.892968878324331, 2.604397960692571, 3.167823504463245, 4.059708610788531, 4.550237596025976, 5.065816854305508, 5.485467963973738, 6.060237444281561, 6.813365062908112, 7.394747829579933, 7.814698207302899, 8.347173129953840, 8.681702274338670, 9.184580062174399, 10.09144001835740, 10.49521140551876, 10.87969114182495, 11.20405258338808, 11.80221321513061, 12.51752536412128, 12.77173676356873, 13.30317404155437, 13.95689580900155

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