Properties

Label 2-379050-1.1-c1-0-0
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 − 2·13-s + 14-s + 16-s + 17-s − 18-s + 21-s − 3·22-s − 8·23-s + 24-s + 2·26-s − 27-s − 28-s + 8·29-s + 2·31-s − 32-s − 3·33-s − 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 − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.218·21-s − 0.639·22-s − 1.66·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.188·28-s + 1.48·29-s + 0.359·31-s − 0.176·32-s − 0.522·33-s − 0.171·34-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: $\chi_{379050} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 379050,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3174607055\)
\(L(\frac12)\) \(\approx\) \(0.3174607055\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−12.20835880061570, −12.02099436423270, −11.71496312073335, −11.11863966525999, −10.45268545506885, −10.18882832804406, −9.952212278059488, −9.333443854040421, −8.858189807129864, −8.477893035084035, −7.893742551915822, −7.431277678261497, −6.936609057899964, −6.469252161778341, −6.179200285228621, −5.586615054776184, −5.114115219853070, −4.351237945703037, −4.076082074857946, −3.377151019913418, −2.783561442872357, −2.219091522081299, −1.486927684402323, −1.111029080693743, −0.1818533451635007, 0.1818533451635007, 1.111029080693743, 1.486927684402323, 2.219091522081299, 2.783561442872357, 3.377151019913418, 4.076082074857946, 4.351237945703037, 5.114115219853070, 5.586615054776184, 6.179200285228621, 6.469252161778341, 6.936609057899964, 7.431277678261497, 7.893742551915822, 8.477893035084035, 8.858189807129864, 9.333443854040421, 9.952212278059488, 10.18882832804406, 10.45268545506885, 11.11863966525999, 11.71496312073335, 12.02099436423270, 12.20835880061570

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