Properties

Label 2-24990-1.1-c1-0-27
Degree $2$
Conductor $24990$
Sign $-1$
Analytic cond. $199.546$
Root an. cond. $14.1260$
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 − 5-s + 6-s − 8-s + 9-s + 10-s − 12-s + 15-s + 16-s + 17-s − 18-s − 2·19-s − 20-s + 24-s + 25-s − 27-s + 4·29-s − 30-s − 8·31-s − 32-s − 34-s + 36-s − 2·37-s + 2·38-s + 40-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.742·29-s − 0.182·30-s − 1.43·31-s − 0.176·32-s − 0.171·34-s + 1/6·36-s − 0.328·37-s + 0.324·38-s + 0.158·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24990\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(199.546\)
Root analytic conductor: \(14.1260\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{24990} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 24990,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 \)
17 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−15.89276683211038, −15.14850592236136, −14.73721457364061, −14.18238634033654, −13.29309444215880, −12.90240815052926, −12.15391212480989, −11.89233727496954, −11.14400122629828, −10.84223453526841, −10.17235923144179, −9.716826575849604, −9.013834495299792, −8.423391260160898, −7.956977002722391, −7.212428357711490, −6.793530602837545, −6.191190203635170, −5.403139475518122, −4.937831741941744, −3.993287757287928, −3.489486529324676, −2.528406687084007, −1.753326475733632, −0.8614831159411040, 0, 0.8614831159411040, 1.753326475733632, 2.528406687084007, 3.489486529324676, 3.993287757287928, 4.937831741941744, 5.403139475518122, 6.191190203635170, 6.793530602837545, 7.212428357711490, 7.956977002722391, 8.423391260160898, 9.013834495299792, 9.716826575849604, 10.17235923144179, 10.84223453526841, 11.14400122629828, 11.89233727496954, 12.15391212480989, 12.90240815052926, 13.29309444215880, 14.18238634033654, 14.73721457364061, 15.14850592236136, 15.89276683211038

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