Properties

Label 2-236992-1.1-c1-0-23
Degree $2$
Conductor $236992$
Sign $1$
Analytic cond. $1892.39$
Root an. cond. $43.5016$
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  − 2·3-s + 7-s + 9-s + 4·13-s − 6·17-s − 2·19-s − 2·21-s − 5·25-s + 4·27-s + 6·29-s + 4·31-s + 2·37-s − 8·39-s + 6·41-s − 8·43-s + 12·47-s + 49-s + 12·51-s + 6·53-s + 4·57-s − 6·59-s + 8·61-s + 63-s + 4·67-s + 2·73-s + 10·75-s + 8·79-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.10·13-s − 1.45·17-s − 0.458·19-s − 0.436·21-s − 25-s + 0.769·27-s + 1.11·29-s + 0.718·31-s + 0.328·37-s − 1.28·39-s + 0.937·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s + 1.68·51-s + 0.824·53-s + 0.529·57-s − 0.781·59-s + 1.02·61-s + 0.125·63-s + 0.488·67-s + 0.234·73-s + 1.15·75-s + 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(236992\)    =    \(2^{6} \cdot 7 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1892.39\)
Root analytic conductor: \(43.5016\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{236992} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 236992,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.752797602\)
\(L(\frac12)\) \(\approx\) \(1.752797602\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
23 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−12.83716939489419, −12.30197941155099, −11.89975363726127, −11.42881012474088, −11.15895739810962, −10.60719653117612, −10.41029142419922, −9.744927334784276, −9.032449702968612, −8.738196690074722, −8.200925857554844, −7.793060316870443, −6.973392114943597, −6.607814766669401, −6.151638052500941, −5.861035017578884, −5.147155631390567, −4.779924787395413, −4.100594714313666, −3.893503986547597, −2.903419647377415, −2.353809610018927, −1.741748620469267, −0.9010821052686619, −0.4974584072505260, 0.4974584072505260, 0.9010821052686619, 1.741748620469267, 2.353809610018927, 2.903419647377415, 3.893503986547597, 4.100594714313666, 4.779924787395413, 5.147155631390567, 5.861035017578884, 6.151638052500941, 6.607814766669401, 6.973392114943597, 7.793060316870443, 8.200925857554844, 8.738196690074722, 9.032449702968612, 9.744927334784276, 10.41029142419922, 10.60719653117612, 11.15895739810962, 11.42881012474088, 11.89975363726127, 12.30197941155099, 12.83716939489419

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