Properties

Label 2-378-1.1-c1-0-6
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 − 3·5-s + 7-s − 8-s + 3·10-s + 3·11-s − 4·13-s − 14-s + 16-s − 6·17-s − 7·19-s − 3·20-s − 3·22-s − 3·23-s + 4·25-s + 4·26-s + 28-s + 5·31-s − 32-s + 6·34-s − 3·35-s − 7·37-s + 7·38-s + 3·40-s − 9·41-s − 10·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s − 0.353·8-s + 0.948·10-s + 0.904·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 1.60·19-s − 0.670·20-s − 0.639·22-s − 0.625·23-s + 4/5·25-s + 0.784·26-s + 0.188·28-s + 0.898·31-s − 0.176·32-s + 1.02·34-s − 0.507·35-s − 1.15·37-s + 1.13·38-s + 0.474·40-s − 1.40·41-s − 1.52·43-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: Trivial
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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−10.92490183559974483870177287472, −10.05189465182372194736077139645, −8.751755972478050515328668545232, −8.335493328080316690083698704832, −7.19284214678952744065952854249, −6.52958134533989476002257403650, −4.74118220843440043197796971108, −3.84049559583729782963394147722, −2.14382090414345698958299236646, 0, 2.14382090414345698958299236646, 3.84049559583729782963394147722, 4.74118220843440043197796971108, 6.52958134533989476002257403650, 7.19284214678952744065952854249, 8.335493328080316690083698704832, 8.751755972478050515328668545232, 10.05189465182372194736077139645, 10.92490183559974483870177287472

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