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Properties

Label 2-379050-1.1-c1-0-140
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 $0$

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 + 4·13-s + 14-s + 16-s + 2·17-s + 18-s + 21-s − 3·22-s + 8·23-s + 24-s + 4·26-s + 27-s + 28-s + 2·29-s + 5·31-s + 32-s − 3·33-s + 2·34-s + 36-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 + 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.218·21-s − 0.639·22-s + 1.66·23-s + 0.204·24-s + 0.784·26-s + 0.192·27-s + 0.188·28-s + 0.371·29-s + 0.898·31-s + 0.176·32-s − 0.522·33-s + 0.342·34-s + 1/6·36-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 379050,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.848869284\)
\(L(\frac12)\) \(\approx\) \(8.848869284\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−12.66488456053729, −12.07980446885592, −11.56361040702247, −11.20702994927230, −10.68256062129184, −10.30684513370550, −9.943046965909941, −9.162464836183056, −8.756838336386013, −8.412599885381767, −7.860325148505557, −7.432288467984787, −7.006283105026737, −6.419861410634204, −5.951014354223092, −5.270029734309388, −5.147705581653552, −4.296538869965460, −4.111931610766933, −3.270310371762024, −2.993885615706042, −2.498535439523378, −1.835821644034767, −1.138477262711153, −0.7047364151280316, 0.7047364151280316, 1.138477262711153, 1.835821644034767, 2.498535439523378, 2.993885615706042, 3.270310371762024, 4.111931610766933, 4.296538869965460, 5.147705581653552, 5.270029734309388, 5.951014354223092, 6.419861410634204, 7.006283105026737, 7.432288467984787, 7.860325148505557, 8.412599885381767, 8.756838336386013, 9.162464836183056, 9.943046965909941, 10.30684513370550, 10.68256062129184, 11.20702994927230, 11.56361040702247, 12.07980446885592, 12.66488456053729

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