Properties

Label 2-379050-1.1-c1-0-89
Degree $2$
Conductor $379050$
Sign $-1$
Analytic cond. $3026.72$
Root an. cond. $55.0157$
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 + 13-s + 14-s + 16-s − 18-s + 21-s + 3·22-s − 7·23-s + 24-s − 26-s − 27-s − 28-s + 4·29-s − 7·31-s − 32-s + 3·33-s + 36-s + 4·37-s − 39-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 + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.218·21-s + 0.639·22-s − 1.45·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.742·29-s − 1.25·31-s − 0.176·32-s + 0.522·33-s + 1/6·36-s + 0.657·37-s − 0.160·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(379050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(3026.72\)
Root analytic conductor: \(55.0157\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{379050} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 379050,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
19 \( 1 \)
good11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−12.46405285124509, −12.25845667292693, −11.84656415840430, −11.15675383776175, −10.82650719467325, −10.50693270742340, −10.04360461204860, −9.511722354365454, −9.269165839853911, −8.528415037894142, −8.147984122855463, −7.703157559582226, −7.297716848416997, −6.624188103698496, −6.385586252779724, −5.680811890803571, −5.475292319423966, −4.851569215597387, −4.084988458721117, −3.798451855811453, −3.019268608784466, −2.474919115575317, −1.976883919742489, −1.287139460118266, −0.5712621503573287, 0, 0.5712621503573287, 1.287139460118266, 1.976883919742489, 2.474919115575317, 3.019268608784466, 3.798451855811453, 4.084988458721117, 4.851569215597387, 5.475292319423966, 5.680811890803571, 6.385586252779724, 6.624188103698496, 7.297716848416997, 7.703157559582226, 8.147984122855463, 8.528415037894142, 9.269165839853911, 9.511722354365454, 10.04360461204860, 10.50693270742340, 10.82650719467325, 11.15675383776175, 11.84656415840430, 12.25845667292693, 12.46405285124509

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