Properties

Label 2-76050-1.1-c1-0-71
Degree $2$
Conductor $76050$
Sign $1$
Analytic cond. $607.262$
Root an. cond. $24.6426$
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-s + 4-s + 2·7-s − 8-s + 4·11-s − 2·14-s + 16-s + 8·17-s + 6·19-s − 4·22-s + 6·23-s + 2·28-s + 4·29-s − 32-s − 8·34-s − 2·37-s − 6·38-s − 2·41-s + 4·43-s + 4·44-s − 6·46-s − 3·49-s − 10·53-s − 2·56-s − 4·58-s + 4·59-s − 10·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 1.20·11-s − 0.534·14-s + 1/4·16-s + 1.94·17-s + 1.37·19-s − 0.852·22-s + 1.25·23-s + 0.377·28-s + 0.742·29-s − 0.176·32-s − 1.37·34-s − 0.328·37-s − 0.973·38-s − 0.312·41-s + 0.609·43-s + 0.603·44-s − 0.884·46-s − 3/7·49-s − 1.37·53-s − 0.267·56-s − 0.525·58-s + 0.520·59-s − 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76050\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(607.262\)
Root analytic conductor: \(24.6426\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{76050} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 76050,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.025585433\)
\(L(\frac12)\) \(\approx\) \(3.025585433\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−14.16949742304532, −13.85678252904631, −12.85010454022209, −12.43709332736075, −11.94561448836546, −11.41186858948533, −11.24251428616080, −10.43525238900811, −9.904652775996557, −9.582219279130410, −8.955506301308104, −8.537715792457992, −7.875731461755266, −7.484208032753178, −7.028602314520817, −6.354055641990871, −5.763971366014735, −5.171480481039022, −4.699201835671582, −3.824627494441918, −3.207166606411550, −2.811239610087525, −1.599228495536421, −1.323380447167499, −0.7265403665557169, 0.7265403665557169, 1.323380447167499, 1.599228495536421, 2.811239610087525, 3.207166606411550, 3.824627494441918, 4.699201835671582, 5.171480481039022, 5.763971366014735, 6.354055641990871, 7.028602314520817, 7.484208032753178, 7.875731461755266, 8.537715792457992, 8.955506301308104, 9.582219279130410, 9.904652775996557, 10.43525238900811, 11.24251428616080, 11.41186858948533, 11.94561448836546, 12.43709332736075, 12.85010454022209, 13.85678252904631, 14.16949742304532

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