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Properties

Label 2-177450-1.1-c1-0-223
Degree $2$
Conductor $177450$
Sign $-1$
Analytic cond. $1416.94$
Root an. cond. $37.6423$
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 + 7-s + 8-s + 9-s + 3·11-s − 12-s + 14-s + 16-s + 7·17-s + 18-s − 5·19-s − 21-s + 3·22-s − 6·23-s − 24-s − 27-s + 28-s + 5·29-s − 2·31-s + 32-s − 3·33-s + 7·34-s + 36-s + 2·37-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 + 0.904·11-s − 0.288·12-s + 0.267·14-s + 1/4·16-s + 1.69·17-s + 0.235·18-s − 1.14·19-s − 0.218·21-s + 0.639·22-s − 1.25·23-s − 0.204·24-s − 0.192·27-s + 0.188·28-s + 0.928·29-s − 0.359·31-s + 0.176·32-s − 0.522·33-s + 1.20·34-s + 1/6·36-s + 0.328·37-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: $-1$
Analytic conductor: \(1416.94\)
Root analytic conductor: \(37.6423\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
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 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.34050383255738, −12.91944638167078, −12.25747122040174, −12.06784944238501, −11.75765514823241, −11.16040871212732, −10.57418199867928, −10.29974667625797, −9.717078081964452, −9.258715098801778, −8.431900190752274, −8.112394474842287, −7.609102172746554, −6.929818771133401, −6.498481120312196, −6.077953966095712, −5.547446171555130, −5.113760495739546, −4.432845161353199, −4.062010753763349, −3.538232021333273, −2.898337558564891, −2.109449720399444, −1.534657253514542, −0.9926222843309958, 0, 0.9926222843309958, 1.534657253514542, 2.109449720399444, 2.898337558564891, 3.538232021333273, 4.062010753763349, 4.432845161353199, 5.113760495739546, 5.547446171555130, 6.077953966095712, 6.498481120312196, 6.929818771133401, 7.609102172746554, 8.112394474842287, 8.431900190752274, 9.258715098801778, 9.717078081964452, 10.29974667625797, 10.57418199867928, 11.16040871212732, 11.75765514823241, 12.06784944238501, 12.25747122040174, 12.91944638167078, 13.34050383255738

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