Properties

Label 2-51842-1.1-c1-0-8
Degree $2$
Conductor $51842$
Sign $-1$
Analytic cond. $413.960$
Root an. cond. $20.3460$
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-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s − 4·11-s − 2·12-s + 16-s + 6·17-s − 18-s − 6·19-s + 4·22-s + 2·24-s − 5·25-s + 4·27-s + 10·29-s − 4·31-s − 32-s + 8·33-s − 6·34-s + 36-s + 2·37-s + 6·38-s + 10·41-s + 4·43-s − 4·44-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.577·12-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 1.37·19-s + 0.852·22-s + 0.408·24-s − 25-s + 0.769·27-s + 1.85·29-s − 0.718·31-s − 0.176·32-s + 1.39·33-s − 1.02·34-s + 1/6·36-s + 0.328·37-s + 0.973·38-s + 1.56·41-s + 0.609·43-s − 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51842\)    =    \(2 \cdot 7^{2} \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(413.960\)
Root analytic conductor: \(20.3460\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{51842} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 51842,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 \)
23 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−14.81571557318654, −14.31335728427563, −13.71849204895443, −12.92260136984502, −12.53031177781911, −12.20332361592663, −11.46765779014662, −11.13703251478352, −10.57525761394699, −10.13629072447967, −9.843436736458840, −9.005757289350659, −8.309523627842303, −8.011975447433408, −7.411380901946456, −6.753478270101633, −6.131942894743525, −5.761058112041112, −5.226504209956532, −4.583673345243754, −3.869926415833682, −2.898059288679726, −2.472996052957225, −1.498267036194116, −0.7059137775649130, 0, 0.7059137775649130, 1.498267036194116, 2.472996052957225, 2.898059288679726, 3.869926415833682, 4.583673345243754, 5.226504209956532, 5.761058112041112, 6.131942894743525, 6.753478270101633, 7.411380901946456, 8.011975447433408, 8.309523627842303, 9.005757289350659, 9.843436736458840, 10.13629072447967, 10.57525761394699, 11.13703251478352, 11.46765779014662, 12.20332361592663, 12.53031177781911, 12.92260136984502, 13.71849204895443, 14.31335728427563, 14.81571557318654

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