Properties

Label 2-232050-1.1-c1-0-9
Degree $2$
Conductor $232050$
Sign $1$
Analytic cond. $1852.92$
Root an. cond. $43.0456$
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 − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 4·11-s − 12-s − 13-s − 14-s + 16-s − 17-s − 18-s + 4·19-s − 21-s + 4·22-s − 8·23-s + 24-s + 26-s − 27-s + 28-s − 2·29-s − 32-s + 4·33-s + 34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.218·21-s + 0.852·22-s − 1.66·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.188·28-s − 0.371·29-s − 0.176·32-s + 0.696·33-s + 0.171·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(1852.92\)
Root analytic conductor: \(43.0456\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 232050,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6268132378\)
\(L(\frac12)\) \(\approx\) \(0.6268132378\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.82345679674187, −12.32683686361966, −11.87313012179426, −11.55062099689255, −10.96564664426636, −10.56195754416871, −10.11923820231872, −9.874403275265632, −9.035715707031412, −8.893629562815060, −8.008791666523778, −7.718162457232222, −7.455757369371641, −6.829146233318977, −6.173886672835227, −5.691496470541413, −5.379429700050559, −4.749242053661936, −4.170562714264310, −3.585373071271735, −2.791248579321449, −2.312424738524693, −1.758970218540645, −1.010284616605529, −0.2860175941342883, 0.2860175941342883, 1.010284616605529, 1.758970218540645, 2.312424738524693, 2.791248579321449, 3.585373071271735, 4.170562714264310, 4.749242053661936, 5.379429700050559, 5.691496470541413, 6.173886672835227, 6.829146233318977, 7.455757369371641, 7.718162457232222, 8.008791666523778, 8.893629562815060, 9.035715707031412, 9.874403275265632, 10.11923820231872, 10.56195754416871, 10.96564664426636, 11.55062099689255, 11.87313012179426, 12.32683686361966, 12.82345679674187

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