Properties

Label 2-93600-1.1-c1-0-21
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  + 6·11-s − 13-s + 4·17-s − 6·19-s − 6·23-s + 2·29-s + 6·37-s − 2·41-s − 12·43-s − 12·47-s − 7·49-s + 10·53-s + 6·59-s + 10·61-s − 12·67-s + 12·71-s − 2·73-s − 12·83-s + 10·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.80·11-s − 0.277·13-s + 0.970·17-s − 1.37·19-s − 1.25·23-s + 0.371·29-s + 0.986·37-s − 0.312·41-s − 1.82·43-s − 1.75·47-s − 49-s + 1.37·53-s + 0.781·59-s + 1.28·61-s − 1.46·67-s + 1.42·71-s − 0.234·73-s − 1.31·83-s + 1.05·89-s + 1.01·97-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 93600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.204684403\)
\(L(\frac12)\) \(\approx\) \(2.204684403\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.93360892783689, −13.23059299337083, −12.92904867482809, −12.20608219920383, −11.81353270341576, −11.58317825368790, −10.93742007212447, −10.10316722411367, −9.974751314007399, −9.446890162210660, −8.756094849412034, −8.296106989952721, −7.978731172155133, −7.096697005435101, −6.651750641067167, −6.277773684990914, −5.738019446686101, −4.963571091450835, −4.437485869082863, −3.803826497891509, −3.497142639612768, −2.605432132848699, −1.844183799486787, −1.395017476334462, −0.4751554220089831, 0.4751554220089831, 1.395017476334462, 1.844183799486787, 2.605432132848699, 3.497142639612768, 3.803826497891509, 4.437485869082863, 4.963571091450835, 5.738019446686101, 6.277773684990914, 6.651750641067167, 7.096697005435101, 7.978731172155133, 8.296106989952721, 8.756094849412034, 9.446890162210660, 9.974751314007399, 10.10316722411367, 10.93742007212447, 11.58317825368790, 11.81353270341576, 12.20608219920383, 12.92904867482809, 13.23059299337083, 13.93360892783689

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