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Properties

Label 2-66248-1.1-c1-0-14
Degree $2$
Conductor $66248$
Sign $-1$
Analytic cond. $528.992$
Root an. cond. $22.9998$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·5-s + 9-s − 6·15-s + 2·17-s + 5·19-s + 23-s + 4·25-s + 4·27-s − 5·29-s − 3·31-s + 6·37-s − 2·41-s + 43-s + 3·45-s + 3·47-s − 4·51-s + 11·53-s − 10·57-s − 8·59-s + 10·61-s − 2·67-s − 2·69-s − 14·71-s − 7·73-s − 8·75-s + 13·79-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.34·5-s + 1/3·9-s − 1.54·15-s + 0.485·17-s + 1.14·19-s + 0.208·23-s + 4/5·25-s + 0.769·27-s − 0.928·29-s − 0.538·31-s + 0.986·37-s − 0.312·41-s + 0.152·43-s + 0.447·45-s + 0.437·47-s − 0.560·51-s + 1.51·53-s − 1.32·57-s − 1.04·59-s + 1.28·61-s − 0.244·67-s − 0.240·69-s − 1.66·71-s − 0.819·73-s − 0.923·75-s + 1.46·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66248\)    =    \(2^{3} \cdot 7^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(528.992\)
Root analytic conductor: \(22.9998\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 66248,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−14.54955371910319, −13.78848945323568, −13.43772749280616, −13.02194351326389, −12.32808189089218, −11.94474830011769, −11.42560744719376, −10.91237449037650, −10.42085484468003, −9.969018975101378, −9.394482806184929, −9.104595506931044, −8.300031564553855, −7.594675698571329, −7.058178085629974, −6.513637084749466, −5.872475946551863, −5.499409521308233, −5.328535250982297, −4.502216164304118, −3.777965183590235, −2.933951733703051, −2.390160521768176, −1.477271620906357, −1.020300684852413, 0, 1.020300684852413, 1.477271620906357, 2.390160521768176, 2.933951733703051, 3.777965183590235, 4.502216164304118, 5.328535250982297, 5.499409521308233, 5.872475946551863, 6.513637084749466, 7.058178085629974, 7.594675698571329, 8.300031564553855, 9.104595506931044, 9.394482806184929, 9.969018975101378, 10.42085484468003, 10.91237449037650, 11.42560744719376, 11.94474830011769, 12.32808189089218, 13.02194351326389, 13.43772749280616, 13.78848945323568, 14.54955371910319

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