Properties

Label 2-93600-1.1-c1-0-18
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  − 2·7-s − 4·11-s + 13-s − 4·17-s + 2·19-s + 8·23-s − 2·31-s + 10·37-s + 10·41-s − 3·49-s − 8·53-s + 4·59-s + 2·61-s − 2·67-s − 2·73-s + 8·77-s + 12·83-s + 6·89-s − 2·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 8·119-s + ⋯
L(s)  = 1  − 0.755·7-s − 1.20·11-s + 0.277·13-s − 0.970·17-s + 0.458·19-s + 1.66·23-s − 0.359·31-s + 1.64·37-s + 1.56·41-s − 3/7·49-s − 1.09·53-s + 0.520·59-s + 0.256·61-s − 0.244·67-s − 0.234·73-s + 0.911·77-s + 1.31·83-s + 0.635·89-s − 0.209·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.733·119-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

\(L(1)\) \(\approx\) \(1.669204178\)
\(L(\frac12)\) \(\approx\) \(1.669204178\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.57878336516048, −13.24619278054160, −12.87834492806575, −12.66921879708226, −11.76957863029562, −11.28945613051895, −10.84019899314324, −10.50326096320183, −9.771362323137270, −9.306099284556949, −9.006123663852269, −8.285174670741109, −7.687325432934008, −7.364012697023383, −6.636110450079199, −6.225662299748320, −5.656878412769937, −4.968441740424268, −4.619934903400670, −3.824733740898902, −3.169527120459890, −2.694298613196042, −2.170774701064016, −1.124294336541760, −0.4496115012329545, 0.4496115012329545, 1.124294336541760, 2.170774701064016, 2.694298613196042, 3.169527120459890, 3.824733740898902, 4.619934903400670, 4.968441740424268, 5.656878412769937, 6.225662299748320, 6.636110450079199, 7.364012697023383, 7.687325432934008, 8.285174670741109, 9.006123663852269, 9.306099284556949, 9.771362323137270, 10.50326096320183, 10.84019899314324, 11.28945613051895, 11.76957863029562, 12.66921879708226, 12.87834492806575, 13.24619278054160, 13.57878336516048

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