Properties

Label 2-235200-1.1-c1-0-432
Degree $2$
Conductor $235200$
Sign $1$
Analytic cond. $1878.08$
Root an. cond. $43.3368$
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-s + 9-s − 5·11-s + 2·13-s + 6·17-s + 2·19-s + 5·23-s + 27-s + 5·29-s − 4·31-s − 5·33-s + 37-s + 2·39-s + 12·41-s − 5·43-s − 2·47-s + 6·51-s + 14·53-s + 2·57-s − 2·59-s + 5·67-s + 5·69-s + 9·71-s + 10·73-s − 11·79-s + 81-s + 16·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.50·11-s + 0.554·13-s + 1.45·17-s + 0.458·19-s + 1.04·23-s + 0.192·27-s + 0.928·29-s − 0.718·31-s − 0.870·33-s + 0.164·37-s + 0.320·39-s + 1.87·41-s − 0.762·43-s − 0.291·47-s + 0.840·51-s + 1.92·53-s + 0.264·57-s − 0.260·59-s + 0.610·67-s + 0.601·69-s + 1.06·71-s + 1.17·73-s − 1.23·79-s + 1/9·81-s + 1.75·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(235200\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1878.08\)
Root analytic conductor: \(43.3368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 235200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.725580226\)
\(L(\frac12)\) \(\approx\) \(4.725580226\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−13.03974210135392, −12.56774074703719, −12.06446718887321, −11.49050360959853, −10.99001306841345, −10.49429171942341, −10.15300239503607, −9.703127830689690, −9.084846677433723, −8.724979741971687, −8.111734581731431, −7.711370253915804, −7.453102644247226, −6.826035927658777, −6.163921564361606, −5.640669237739578, −5.141538734021220, −4.808434718884130, −4.008189182908257, −3.355180381660329, −3.141966125069906, −2.421206903104403, −1.979544384602657, −0.9264519113286336, −0.7329502818576649, 0.7329502818576649, 0.9264519113286336, 1.979544384602657, 2.421206903104403, 3.141966125069906, 3.355180381660329, 4.008189182908257, 4.808434718884130, 5.141538734021220, 5.640669237739578, 6.163921564361606, 6.826035927658777, 7.453102644247226, 7.711370253915804, 8.111734581731431, 8.724979741971687, 9.084846677433723, 9.703127830689690, 10.15300239503607, 10.49429171942341, 10.99001306841345, 11.49050360959853, 12.06446718887321, 12.56774074703719, 13.03974210135392

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