Properties

Label 2-379050-1.1-c1-0-29
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 − 12-s − 3·13-s + 14-s + 16-s + 4·17-s − 18-s + 21-s − 4·23-s + 24-s + 3·26-s − 27-s − 28-s − 10·29-s − 31-s − 32-s − 4·34-s + 36-s + 8·37-s + 3·39-s + 7·41-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 − 0.288·12-s − 0.832·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.218·21-s − 0.834·23-s + 0.204·24-s + 0.588·26-s − 0.192·27-s − 0.188·28-s − 1.85·29-s − 0.179·31-s − 0.176·32-s − 0.685·34-s + 1/6·36-s + 1.31·37-s + 0.480·39-s + 1.09·41-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: \(0\)
Selberg data: \((2,\ 379050,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9569552454\)
\(L(\frac12)\) \(\approx\) \(0.9569552454\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−12.33331283254963, −11.97272646058025, −11.59371903214177, −11.03389200412662, −10.70057777926773, −10.14113764426328, −9.700616003401194, −9.466864061444936, −9.060610405260905, −8.187908598556780, −7.940495492795366, −7.473011713687732, −7.035027127641840, −6.517930820143300, −6.026042733439020, −5.489061402730426, −5.255886665270950, −4.483824149618529, −3.809023872763545, −3.558147592480201, −2.663280010043163, −2.256967688719074, −1.647004540052751, −0.8886051944854910, −0.3601278227512912, 0.3601278227512912, 0.8886051944854910, 1.647004540052751, 2.256967688719074, 2.663280010043163, 3.558147592480201, 3.809023872763545, 4.483824149618529, 5.255886665270950, 5.489061402730426, 6.026042733439020, 6.517930820143300, 7.035027127641840, 7.473011713687732, 7.940495492795366, 8.187908598556780, 9.060610405260905, 9.466864061444936, 9.700616003401194, 10.14113764426328, 10.70057777926773, 11.03389200412662, 11.59371903214177, 11.97272646058025, 12.33331283254963

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