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Properties

Label 2-379050-1.1-c1-0-175
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 + 2·11-s − 12-s − 6·13-s + 14-s + 16-s + 5·17-s − 18-s + 21-s − 2·22-s + 6·23-s + 24-s + 6·26-s − 27-s − 28-s + 8·29-s − 8·31-s − 32-s − 2·33-s − 5·34-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 + 0.603·11-s − 0.288·12-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 1.21·17-s − 0.235·18-s + 0.218·21-s − 0.426·22-s + 1.25·23-s + 0.204·24-s + 1.17·26-s − 0.192·27-s − 0.188·28-s + 1.48·29-s − 1.43·31-s − 0.176·32-s − 0.348·33-s − 0.857·34-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: \(1\)
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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−12.55200330574179, −12.15978496341133, −11.88038020692488, −11.27325843232454, −10.89635659974827, −10.38749772904116, −9.870511204148226, −9.688821823468236, −9.134587854365031, −8.799318506641024, −8.030030671366125, −7.582832930560094, −7.297614447903254, −6.708147505250259, −6.439188294700994, −5.721263990073406, −5.319315054307106, −4.843661582793748, −4.295073481720309, −3.591768480199062, −3.070706695948636, −2.524053392445387, −1.974863270267696, −1.089209518682744, −0.8047575097569395, 0, 0.8047575097569395, 1.089209518682744, 1.974863270267696, 2.524053392445387, 3.070706695948636, 3.591768480199062, 4.295073481720309, 4.843661582793748, 5.319315054307106, 5.721263990073406, 6.439188294700994, 6.708147505250259, 7.297614447903254, 7.582832930560094, 8.030030671366125, 8.799318506641024, 9.134587854365031, 9.688821823468236, 9.870511204148226, 10.38749772904116, 10.89635659974827, 11.27325843232454, 11.88038020692488, 12.15978496341133, 12.55200330574179

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