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Properties

Label 2-379050-1.1-c1-0-11
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 − 13-s + 14-s + 16-s − 18-s − 21-s − 3·22-s + 3·23-s − 24-s + 26-s + 27-s − 28-s − 8·29-s − 9·31-s − 32-s + 3·33-s + 36-s − 8·37-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 + 0.625·23-s − 0.204·24-s + 0.196·26-s + 0.192·27-s − 0.188·28-s − 1.48·29-s − 1.61·31-s − 0.176·32-s + 0.522·33-s + 1/6·36-s − 1.31·37-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 379050,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8688780996\)
\(L(\frac12)\) \(\approx\) \(0.8688780996\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−12.55058166456526, −12.01848585552037, −11.51465318538550, −10.98708360321471, −10.68561086528155, −10.18036257016425, −9.465582465098897, −9.324584085958966, −8.979202930097440, −8.573158187192458, −7.756551779864303, −7.601288585771670, −6.916720071474867, −6.835427775165547, −5.961766989571139, −5.673927599430113, −5.037018571870929, −4.296006602029416, −3.846036809445785, −3.371322552625467, −2.847363864781349, −2.228602574536600, −1.599023201294181, −1.264340255633422, −0.2537691612164213, 0.2537691612164213, 1.264340255633422, 1.599023201294181, 2.228602574536600, 2.847363864781349, 3.371322552625467, 3.846036809445785, 4.296006602029416, 5.037018571870929, 5.673927599430113, 5.961766989571139, 6.835427775165547, 6.916720071474867, 7.601288585771670, 7.756551779864303, 8.573158187192458, 8.979202930097440, 9.324584085958966, 9.465582465098897, 10.18036257016425, 10.68561086528155, 10.98708360321471, 11.51465318538550, 12.01848585552037, 12.55058166456526

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