Properties

Degree $2$
Conductor $177450$
Sign $-1$
Motivic weight $1$
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 + 7-s + 8-s + 9-s − 12-s + 14-s + 16-s + 6·17-s + 18-s + 4·19-s − 21-s − 24-s − 27-s + 28-s − 6·29-s + 4·31-s + 32-s + 6·34-s + 36-s + 2·37-s + 4·38-s − 6·41-s − 42-s − 8·43-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.218·21-s − 0.204·24-s − 0.192·27-s + 0.188·28-s − 1.11·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.328·37-s + 0.648·38-s − 0.937·41-s − 0.154·42-s − 1.21·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{177450} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177450,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
13 \( 1 \)
good11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−13.23957982434998, −13.02956029114208, −12.46907894158106, −11.85943439844901, −11.60447466148275, −11.36710458175457, −10.57932305246922, −10.23394653129571, −9.603504233922508, −9.454823099621902, −8.309172276617504, −8.154051896209749, −7.571779540526922, −7.005129171962144, −6.568113524406891, −6.016840842917124, −5.352844880568423, −5.174732620979290, −4.718430329474392, −3.815113280296829, −3.552322552630517, −2.940398376294641, −2.166904282348720, −1.460329120747594, −1.016886220946540, 0, 1.016886220946540, 1.460329120747594, 2.166904282348720, 2.940398376294641, 3.552322552630517, 3.815113280296829, 4.718430329474392, 5.174732620979290, 5.352844880568423, 6.016840842917124, 6.568113524406891, 7.005129171962144, 7.571779540526922, 8.154051896209749, 8.309172276617504, 9.454823099621902, 9.603504233922508, 10.23394653129571, 10.57932305246922, 11.36710458175457, 11.60447466148275, 11.85943439844901, 12.46907894158106, 13.02956029114208, 13.23957982434998

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