Properties

Label 2-93600-1.1-c1-0-128
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  − 3·7-s + 3·11-s − 13-s − 7·17-s − 2·19-s + 23-s − 6·29-s − 10·31-s − 37-s + 41-s − 4·43-s − 4·47-s + 2·49-s + 53-s + 13·61-s − 8·67-s − 5·71-s − 10·73-s − 9·77-s − 79-s − 6·83-s − 9·89-s + 3·91-s + 97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 1.13·7-s + 0.904·11-s − 0.277·13-s − 1.69·17-s − 0.458·19-s + 0.208·23-s − 1.11·29-s − 1.79·31-s − 0.164·37-s + 0.156·41-s − 0.609·43-s − 0.583·47-s + 2/7·49-s + 0.137·53-s + 1.66·61-s − 0.977·67-s − 0.593·71-s − 1.17·73-s − 1.02·77-s − 0.112·79-s − 0.658·83-s − 0.953·89-s + 0.314·91-s + 0.101·97-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: $\chi_{93600} (1, \cdot )$
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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−14.40828259835849, −13.69874377049013, −13.18997180984091, −12.91214095954482, −12.55754076397329, −11.73450210126054, −11.43599717770412, −10.88704345272466, −10.39021676134074, −9.717797849398318, −9.353323847318580, −8.879859990860667, −8.562899408581669, −7.658104123666399, −7.104185429364936, −6.692477011582633, −6.347516616049421, −5.634721257760045, −5.142261572291142, −4.205476722103716, −4.011589544816043, −3.319670933027645, −2.659135944593553, −1.979724148237976, −1.371764018724948, 0, 0, 1.371764018724948, 1.979724148237976, 2.659135944593553, 3.319670933027645, 4.011589544816043, 4.205476722103716, 5.142261572291142, 5.634721257760045, 6.347516616049421, 6.692477011582633, 7.104185429364936, 7.658104123666399, 8.562899408581669, 8.879859990860667, 9.353323847318580, 9.717797849398318, 10.39021676134074, 10.88704345272466, 11.43599717770412, 11.73450210126054, 12.55754076397329, 12.91214095954482, 13.18997180984091, 13.69874377049013, 14.40828259835849

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