Properties

Label 2-378-1.1-c1-0-3
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 + 5-s − 7-s + 8-s + 10-s + 5·11-s − 14-s + 16-s + 2·17-s − 19-s + 20-s + 5·22-s − 23-s − 4·25-s − 28-s + 4·29-s − 9·31-s + 32-s + 2·34-s − 35-s + 5·37-s − 38-s + 40-s − 9·41-s − 10·43-s + 5·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s + 1.50·11-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.229·19-s + 0.223·20-s + 1.06·22-s − 0.208·23-s − 4/5·25-s − 0.188·28-s + 0.742·29-s − 1.61·31-s + 0.176·32-s + 0.342·34-s − 0.169·35-s + 0.821·37-s − 0.162·38-s + 0.158·40-s − 1.40·41-s − 1.52·43-s + 0.753·44-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

\(L(1)\) \(\approx\) \(2.211658448\)
\(L(\frac12)\) \(\approx\) \(2.211658448\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−11.66207448326126921977789851208, −10.47585711653745414595121031437, −9.614675318763964452394438121737, −8.693710276630941434637211750501, −7.33295252843734147038723504601, −6.41477474003074322919370438778, −5.64016362400921902023132431260, −4.30159478875183316412804698377, −3.30945073751881784724147936210, −1.71709282690249260530994735777, 1.71709282690249260530994735777, 3.30945073751881784724147936210, 4.30159478875183316412804698377, 5.64016362400921902023132431260, 6.41477474003074322919370438778, 7.33295252843734147038723504601, 8.693710276630941434637211750501, 9.614675318763964452394438121737, 10.47585711653745414595121031437, 11.66207448326126921977789851208

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