Properties

Label 2-236992-1.1-c1-0-1
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 − 2·5-s − 7-s + 9-s + 2·13-s − 4·15-s + 2·17-s − 4·19-s − 2·21-s − 25-s − 4·27-s + 2·29-s − 10·31-s + 2·35-s + 8·37-s + 4·39-s + 2·41-s − 4·43-s − 2·45-s − 6·47-s + 49-s + 4·51-s − 12·53-s − 8·57-s − 10·59-s − 6·61-s − 63-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s − 1.03·15-s + 0.485·17-s − 0.917·19-s − 0.436·21-s − 1/5·25-s − 0.769·27-s + 0.371·29-s − 1.79·31-s + 0.338·35-s + 1.31·37-s + 0.640·39-s + 0.312·41-s − 0.609·43-s − 0.298·45-s − 0.875·47-s + 1/7·49-s + 0.560·51-s − 1.64·53-s − 1.05·57-s − 1.30·59-s − 0.768·61-s − 0.125·63-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

\(L(1)\) \(\approx\) \(0.5841524931\)
\(L(\frac12)\) \(\approx\) \(0.5841524931\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−12.85058764447447, −12.64204657385624, −11.99511212138191, −11.46448058346217, −11.09375665674910, −10.63335382298224, −10.00070761290611, −9.539282707850460, −8.955537217019504, −8.827392687569129, −8.095485334334289, −7.719707343857871, −7.587832931200599, −6.779673855385644, −6.193528588745836, −5.879189076230113, −5.071439364793265, −4.421349361367768, −4.011921208306032, −3.446183751740447, −3.146301171416020, −2.566892500708475, −1.816060222823631, −1.333538967704276, −0.1866684238585408, 0.1866684238585408, 1.333538967704276, 1.816060222823631, 2.566892500708475, 3.146301171416020, 3.446183751740447, 4.011921208306032, 4.421349361367768, 5.071439364793265, 5.879189076230113, 6.193528588745836, 6.779673855385644, 7.587832931200599, 7.719707343857871, 8.095485334334289, 8.827392687569129, 8.955537217019504, 9.539282707850460, 10.00070761290611, 10.63335382298224, 11.09375665674910, 11.46448058346217, 11.99511212138191, 12.64204657385624, 12.85058764447447

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