Properties

Label 2-93600-1.1-c1-0-77
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 + 13-s + 3·19-s + 4·23-s + 29-s + 8·31-s + 37-s − 41-s + 6·43-s + 11·47-s + 9·49-s − 3·53-s + 10·59-s + 4·61-s + 13·67-s + 9·71-s + 3·79-s − 2·83-s − 10·89-s + 4·91-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.51·7-s + 0.277·13-s + 0.688·19-s + 0.834·23-s + 0.185·29-s + 1.43·31-s + 0.164·37-s − 0.156·41-s + 0.914·43-s + 1.60·47-s + 9/7·49-s − 0.412·53-s + 1.30·59-s + 0.512·61-s + 1.58·67-s + 1.06·71-s + 0.337·79-s − 0.219·83-s − 1.05·89-s + 0.419·91-s + 0.812·97-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

\(L(1)\) \(\approx\) \(4.550070776\)
\(L(\frac12)\) \(\approx\) \(4.550070776\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−13.94233558269590, −13.47662891893998, −12.77299266970862, −12.35064982894474, −11.72803874718559, −11.34630629094900, −11.02641608634684, −10.40389572478739, −9.931130160270584, −9.272864922501530, −8.772695003126832, −8.224063663917346, −7.930993468902883, −7.264469253583800, −6.836985672310381, −6.122909067078706, −5.445136086354128, −5.119542213444373, −4.488803738397935, −4.026226431385478, −3.273583833646157, −2.520194711914288, −2.041322828387786, −1.068604367936621, −0.8399980321543461, 0.8399980321543461, 1.068604367936621, 2.041322828387786, 2.520194711914288, 3.273583833646157, 4.026226431385478, 4.488803738397935, 5.119542213444373, 5.445136086354128, 6.122909067078706, 6.836985672310381, 7.264469253583800, 7.930993468902883, 8.224063663917346, 8.772695003126832, 9.272864922501530, 9.931130160270584, 10.40389572478739, 11.02641608634684, 11.34630629094900, 11.72803874718559, 12.35064982894474, 12.77299266970862, 13.47662891893998, 13.94233558269590

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