Properties

Label 2-378-1.1-c1-0-7
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 $1$

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: \(1\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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

−10.88167975391907609356041649169, −9.967814569800271651158865125350, −9.099693834078022557750942008486, −8.015044568391676191880321656496, −7.43396451799616319962951174764, −6.23227538486956081778076113568, −5.08207234913297047777508467306, −3.57313298834818934312909000630, −2.24049119020186278469176393363, 0, 2.24049119020186278469176393363, 3.57313298834818934312909000630, 5.08207234913297047777508467306, 6.23227538486956081778076113568, 7.43396451799616319962951174764, 8.015044568391676191880321656496, 9.099693834078022557750942008486, 9.967814569800271651158865125350, 10.88167975391907609356041649169

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