Properties

Label 2-247962-1.1-c1-0-26
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 2·5-s + 6-s + 8-s + 9-s + 2·10-s − 11-s + 12-s + 13-s + 2·15-s + 16-s + 18-s + 8·19-s + 2·20-s − 22-s + 4·23-s + 24-s − 25-s + 26-s + 27-s − 6·29-s + 2·30-s + 4·31-s + 32-s − 33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s + 0.277·13-s + 0.516·15-s + 1/4·16-s + 0.235·18-s + 1.83·19-s + 0.447·20-s − 0.213·22-s + 0.834·23-s + 0.204·24-s − 1/5·25-s + 0.196·26-s + 0.192·27-s − 1.11·29-s + 0.365·30-s + 0.718·31-s + 0.176·32-s − 0.174·33-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: \(0\)
Selberg data: \((2,\ 247962,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(10.05698823\)
\(L(\frac12)\) \(\approx\) \(10.05698823\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.03516900073665, −12.54181935492305, −12.08449745196174, −11.35187319359699, −11.17541473576472, −10.60127910638633, −9.938554663447351, −9.619950178167487, −9.259839942655646, −8.769396788726359, −7.939010944425031, −7.722538706193064, −7.155857212716473, −6.723125354648166, −5.927270809786219, −5.712502611982121, −5.267841439332874, −4.627889243052570, −4.063322642908199, −3.540176969205490, −2.923719470725889, −2.513506217993213, −2.017494796160862, −1.193005166801688, −0.7956456734744213, 0.7956456734744213, 1.193005166801688, 2.017494796160862, 2.513506217993213, 2.923719470725889, 3.540176969205490, 4.063322642908199, 4.627889243052570, 5.267841439332874, 5.712502611982121, 5.927270809786219, 6.723125354648166, 7.155857212716473, 7.722538706193064, 7.939010944425031, 8.769396788726359, 9.259839942655646, 9.619950178167487, 9.938554663447351, 10.60127910638633, 11.17541473576472, 11.35187319359699, 12.08449745196174, 12.54181935492305, 13.03516900073665

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