Properties

Label 2-94136-1.1-c1-0-1
Degree $2$
Conductor $94136$
Sign $1$
Analytic cond. $751.679$
Root an. cond. $27.4167$
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  − 3·3-s + 3·5-s + 7-s + 6·9-s + 4·13-s − 9·15-s − 3·17-s + 4·19-s − 3·21-s + 4·23-s + 4·25-s − 9·27-s + 3·29-s + 5·31-s + 3·35-s − 10·37-s − 12·39-s − 5·43-s + 18·45-s − 6·47-s + 49-s + 9·51-s + 3·53-s − 12·57-s + 6·59-s − 3·61-s + 6·63-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.34·5-s + 0.377·7-s + 2·9-s + 1.10·13-s − 2.32·15-s − 0.727·17-s + 0.917·19-s − 0.654·21-s + 0.834·23-s + 4/5·25-s − 1.73·27-s + 0.557·29-s + 0.898·31-s + 0.507·35-s − 1.64·37-s − 1.92·39-s − 0.762·43-s + 2.68·45-s − 0.875·47-s + 1/7·49-s + 1.26·51-s + 0.412·53-s − 1.58·57-s + 0.781·59-s − 0.384·61-s + 0.755·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 94136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(94136\)    =    \(2^{3} \cdot 7 \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(751.679\)
Root analytic conductor: \(27.4167\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 94136,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.084951757\)
\(L(\frac12)\) \(\approx\) \(2.084951757\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
41 \( 1 \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
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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

−13.67163909374911, −13.21596480728899, −13.00009145225136, −12.15075735245395, −11.83641800819399, −11.31639762507645, −10.91704373541281, −10.35851441764913, −10.11190073854576, −9.511592835197540, −8.871136937917399, −8.451922251526735, −7.627848683842845, −6.882620671511134, −6.539128960105439, −6.235334151609977, −5.469314362922917, −5.262906962619443, −4.800459026650839, −4.083438751767809, −3.320784812683206, −2.495922614102425, −1.611576691357941, −1.306236929138777, −0.5508661707446547, 0.5508661707446547, 1.306236929138777, 1.611576691357941, 2.495922614102425, 3.320784812683206, 4.083438751767809, 4.800459026650839, 5.262906962619443, 5.469314362922917, 6.235334151609977, 6.539128960105439, 6.882620671511134, 7.627848683842845, 8.451922251526735, 8.871136937917399, 9.511592835197540, 10.11190073854576, 10.35851441764913, 10.91704373541281, 11.31639762507645, 11.83641800819399, 12.15075735245395, 13.00009145225136, 13.21596480728899, 13.67163909374911

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