Properties

Label 2-378-1.1-c1-0-2
Degree $2$
Conductor $378$
Sign $1$
Analytic cond. $3.01834$
Root an. cond. $1.73733$
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 + 4-s + 7-s + 8-s + 5·13-s + 14-s + 16-s + 3·17-s + 2·19-s − 9·23-s − 5·25-s + 5·26-s + 28-s − 3·29-s + 5·31-s + 32-s + 3·34-s + 2·37-s + 2·38-s − 6·41-s − 43-s − 9·46-s − 6·47-s + 49-s − 5·50-s + 5·52-s + 3·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1.38·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.458·19-s − 1.87·23-s − 25-s + 0.980·26-s + 0.188·28-s − 0.557·29-s + 0.898·31-s + 0.176·32-s + 0.514·34-s + 0.328·37-s + 0.324·38-s − 0.937·41-s − 0.152·43-s − 1.32·46-s − 0.875·47-s + 1/7·49-s − 0.707·50-s + 0.693·52-s + 0.412·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $1$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{378} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.172706901\)
\(L(\frac12)\) \(\approx\) \(2.172706901\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−11.61812839447450755099610871682, −10.58318399981159313328624140160, −9.704000550255874231272189290913, −8.360658565008429029479731162281, −7.66081859530557802153309309893, −6.29670817383847541122613475379, −5.61879984028395623126782300752, −4.30645127647783484429164317499, −3.34773629764867655679280376645, −1.68869994047015348511995782895, 1.68869994047015348511995782895, 3.34773629764867655679280376645, 4.30645127647783484429164317499, 5.61879984028395623126782300752, 6.29670817383847541122613475379, 7.66081859530557802153309309893, 8.360658565008429029479731162281, 9.704000550255874231272189290913, 10.58318399981159313328624140160, 11.61812839447450755099610871682

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