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Properties

Label 2-93600-1.1-c1-0-85
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 + 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: 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 \( 1 - 5 T + p T^{2} \)
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

−14.07289034084385, −13.69940876426907, −13.05737085818610, −12.58197529685315, −12.07737047365010, −11.67860624543814, −11.32726608092735, −10.45973918543522, −9.849796381841401, −9.741522292468688, −9.268059023315053, −8.537532336917554, −8.227225691955893, −7.304699035202273, −6.940770646478783, −6.329973938867424, −6.214970545822723, −5.248968950029702, −4.942452880063986, −3.820120061954793, −3.557466882365552, −3.286184624691926, −2.323920683036090, −1.546431605936014, −0.8906003678543776, 0, 0.8906003678543776, 1.546431605936014, 2.323920683036090, 3.286184624691926, 3.557466882365552, 3.820120061954793, 4.942452880063986, 5.248968950029702, 6.214970545822723, 6.329973938867424, 6.940770646478783, 7.304699035202273, 8.227225691955893, 8.537532336917554, 9.268059023315053, 9.741522292468688, 9.849796381841401, 10.45973918543522, 11.32726608092735, 11.67860624543814, 12.07737047365010, 12.58197529685315, 13.05737085818610, 13.69940876426907, 14.07289034084385

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