Properties

Label 2-43350-1.1-c1-0-70
Degree $2$
Conductor $43350$
Sign $-1$
Analytic cond. $346.151$
Root an. cond. $18.6051$
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 − 4·7-s + 8-s + 9-s + 12-s − 2·13-s − 4·14-s + 16-s + 18-s − 4·19-s − 4·21-s + 24-s − 2·26-s + 27-s − 4·28-s + 6·29-s − 8·31-s + 32-s + 36-s + 2·37-s − 4·38-s − 2·39-s + 6·41-s − 4·42-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.235·18-s − 0.917·19-s − 0.872·21-s + 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.755·28-s + 1.11·29-s − 1.43·31-s + 0.176·32-s + 1/6·36-s + 0.328·37-s − 0.648·38-s − 0.320·39-s + 0.937·41-s − 0.617·42-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(346.151\)
Root analytic conductor: \(18.6051\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 43350,\ (\ :1/2),\ -1)\)

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 \( 1 \)
17 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−14.79790890640232, −14.39701753093196, −14.02487727989374, −13.16063466901853, −12.95779676480166, −12.66329295191447, −11.99650419403939, −11.46601763578711, −10.58019880787853, −10.36357785424678, −9.666425384421297, −9.216074120957454, −8.681992818329682, −7.964823849968578, −7.297150403381770, −6.877957310823315, −6.308262810962315, −5.808182491958193, −5.092318085110258, −4.332896146069050, −3.820862757634631, −3.289692850515834, −2.521690148990271, −2.249564544407232, −1.034045180816660, 0, 1.034045180816660, 2.249564544407232, 2.521690148990271, 3.289692850515834, 3.820862757634631, 4.332896146069050, 5.092318085110258, 5.808182491958193, 6.308262810962315, 6.877957310823315, 7.297150403381770, 7.964823849968578, 8.681992818329682, 9.216074120957454, 9.666425384421297, 10.36357785424678, 10.58019880787853, 11.46601763578711, 11.99650419403939, 12.66329295191447, 12.95779676480166, 13.16063466901853, 14.02487727989374, 14.39701753093196, 14.79790890640232

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