Properties

Label 2-59248-1.1-c1-0-27
Degree $2$
Conductor $59248$
Sign $-1$
Analytic cond. $473.097$
Root an. cond. $21.7508$
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 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 59248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(59248\)    =    \(2^{4} \cdot 7 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(473.097\)
Root analytic conductor: \(21.7508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{59248} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 59248,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
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

−14.39757105143842, −14.16806699041746, −13.55856988089656, −13.36045776365496, −12.69675964577233, −11.94580302379627, −11.78433630825493, −10.84452295633996, −10.26794481577303, −10.02190574024505, −9.363165123910245, −8.852259347707245, −8.489943772692464, −7.902542232144033, −7.404884668191178, −6.778742799958364, −6.089104666235443, −5.598889453313342, −4.977442209389217, −4.261237216372055, −3.726770063975753, −2.863069576436789, −2.499387442230968, −1.882882420766817, −1.243956171174173, 0, 1.243956171174173, 1.882882420766817, 2.499387442230968, 2.863069576436789, 3.726770063975753, 4.261237216372055, 4.977442209389217, 5.598889453313342, 6.089104666235443, 6.778742799958364, 7.404884668191178, 7.902542232144033, 8.489943772692464, 8.852259347707245, 9.363165123910245, 10.02190574024505, 10.26794481577303, 10.84452295633996, 11.78433630825493, 11.94580302379627, 12.69675964577233, 13.36045776365496, 13.55856988089656, 14.16806699041746, 14.39757105143842

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