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Properties

Label 2-124950-1.1-c1-0-133
Degree $2$
Conductor $124950$
Sign $-1$
Analytic cond. $997.730$
Root an. cond. $31.5868$
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 + 6-s − 8-s + 9-s − 2·11-s − 12-s + 16-s + 17-s − 18-s − 19-s + 2·22-s + 24-s − 27-s + 29-s + 8·31-s − 32-s + 2·33-s − 34-s + 36-s − 2·37-s + 38-s − 2·41-s − 4·43-s − 2·44-s + 6·47-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.229·19-s + 0.426·22-s + 0.204·24-s − 0.192·27-s + 0.185·29-s + 1.43·31-s − 0.176·32-s + 0.348·33-s − 0.171·34-s + 1/6·36-s − 0.328·37-s + 0.162·38-s − 0.312·41-s − 0.609·43-s − 0.301·44-s + 0.875·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(997.730\)
Root analytic conductor: \(31.5868\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 124950,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
17 \( 1 - T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
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 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
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

−13.66715241278443, −13.27402881282162, −12.68533174361658, −12.15724362403164, −11.83066548780114, −11.34133705727011, −10.63053505700791, −10.51774627666214, −9.888405672194592, −9.539026339433926, −8.829094774858873, −8.320156048189453, −7.973025740926818, −7.347957539695369, −6.784514634899506, −6.450123709156757, −5.769214710733699, −5.263037456726396, −4.812171455506517, −4.024731834924507, −3.509892503483859, −2.605062160655522, −2.304542082095270, −1.347158416771151, −0.7898287106523867, 0, 0.7898287106523867, 1.347158416771151, 2.304542082095270, 2.605062160655522, 3.509892503483859, 4.024731834924507, 4.812171455506517, 5.263037456726396, 5.769214710733699, 6.450123709156757, 6.784514634899506, 7.347957539695369, 7.973025740926818, 8.320156048189453, 8.829094774858873, 9.539026339433926, 9.888405672194592, 10.51774627666214, 10.63053505700791, 11.34133705727011, 11.83066548780114, 12.15724362403164, 12.68533174361658, 13.27402881282162, 13.66715241278443

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