Properties

Label 2-247962-1.1-c1-0-29
Degree $2$
Conductor $247962$
Sign $-1$
Analytic cond. $1979.98$
Root an. cond. $44.4970$
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 − 3-s + 4-s + 4·5-s + 6-s + 2·7-s − 8-s + 9-s − 4·10-s − 11-s − 12-s − 13-s − 2·14-s − 4·15-s + 16-s − 18-s − 4·19-s + 4·20-s − 2·21-s + 22-s + 6·23-s + 24-s + 11·25-s + 26-s − 27-s + 2·28-s + 4·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.534·14-s − 1.03·15-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.894·20-s − 0.436·21-s + 0.213·22-s + 1.25·23-s + 0.204·24-s + 11/5·25-s + 0.196·26-s − 0.192·27-s + 0.377·28-s + 0.730·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 247962 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247962 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(247962\)    =    \(2 \cdot 3 \cdot 11 \cdot 13 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(1979.98\)
Root analytic conductor: \(44.4970\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{247962} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 247962,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 \)
good5 \( 1 - 4 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
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

−13.04852730965123, −12.69991670512068, −12.15761906653605, −11.53361363911899, −11.08092164294774, −10.60024129199398, −10.46427814076613, −9.780652489586036, −9.508916493881538, −8.956724140330387, −8.418050821483188, −8.195146049397138, −7.201654765164786, −6.843241569469194, −6.677569724778581, −5.854753129592623, −5.439645568862466, −5.068275731553697, −4.763961962968830, −3.691239620779929, −3.190904186008815, −2.275440327919508, −1.963760141607562, −1.591064257294351, −0.8504350381035540, 0, 0.8504350381035540, 1.591064257294351, 1.963760141607562, 2.275440327919508, 3.190904186008815, 3.691239620779929, 4.763961962968830, 5.068275731553697, 5.439645568862466, 5.854753129592623, 6.677569724778581, 6.843241569469194, 7.201654765164786, 8.195146049397138, 8.418050821483188, 8.956724140330387, 9.508916493881538, 9.780652489586036, 10.46427814076613, 10.60024129199398, 11.08092164294774, 11.53361363911899, 12.15761906653605, 12.69991670512068, 13.04852730965123

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