Properties

Label 2-378-1.1-c3-0-2
Degree $2$
Conductor $378$
Sign $1$
Analytic cond. $22.3027$
Root an. cond. $4.72257$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s + 7·5-s − 7·7-s − 8·8-s − 14·10-s − 28·11-s + 30·13-s + 14·14-s + 16·16-s + 47·17-s + 164·19-s + 28·20-s + 56·22-s − 94·23-s − 76·25-s − 60·26-s − 28·28-s − 200·29-s + 162·31-s − 32·32-s − 94·34-s − 49·35-s + 137·37-s − 328·38-s − 56·40-s + 141·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.626·5-s − 0.377·7-s − 0.353·8-s − 0.442·10-s − 0.767·11-s + 0.640·13-s + 0.267·14-s + 1/4·16-s + 0.670·17-s + 1.98·19-s + 0.313·20-s + 0.542·22-s − 0.852·23-s − 0.607·25-s − 0.452·26-s − 0.188·28-s − 1.28·29-s + 0.938·31-s − 0.176·32-s − 0.474·34-s − 0.236·35-s + 0.608·37-s − 1.40·38-s − 0.221·40-s + 0.537·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $1$
Analytic conductor: \(22.3027\)
Root analytic conductor: \(4.72257\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.450275127\)
\(L(\frac12)\) \(\approx\) \(1.450275127\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
3 \( 1 \)
7 \( 1 + p T \)
good5 \( 1 - 7 T + p^{3} T^{2} \)
11 \( 1 + 28 T + p^{3} T^{2} \)
13 \( 1 - 30 T + p^{3} T^{2} \)
17 \( 1 - 47 T + p^{3} T^{2} \)
19 \( 1 - 164 T + p^{3} T^{2} \)
23 \( 1 + 94 T + p^{3} T^{2} \)
29 \( 1 + 200 T + p^{3} T^{2} \)
31 \( 1 - 162 T + p^{3} T^{2} \)
37 \( 1 - 137 T + p^{3} T^{2} \)
41 \( 1 - 141 T + p^{3} T^{2} \)
43 \( 1 - 293 T + p^{3} T^{2} \)
47 \( 1 - 471 T + p^{3} T^{2} \)
53 \( 1 + 306 T + p^{3} T^{2} \)
59 \( 1 - 331 T + p^{3} T^{2} \)
61 \( 1 + 204 T + p^{3} T^{2} \)
67 \( 1 - 928 T + p^{3} T^{2} \)
71 \( 1 - 740 T + p^{3} T^{2} \)
73 \( 1 - 706 T + p^{3} T^{2} \)
79 \( 1 + 195 T + p^{3} T^{2} \)
83 \( 1 - 485 T + p^{3} T^{2} \)
89 \( 1 - 114 T + p^{3} T^{2} \)
97 \( 1 + 344 T + p^{3} T^{2} \)
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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

−10.76964383637475038497786200999, −9.777227375654841080951682406845, −9.422397525258518606352803682510, −8.071979864346530587744308799340, −7.41630612774521768145736870279, −6.08387929336352569408629941974, −5.42182873989330627685339758405, −3.61601797549595869986265934658, −2.36557004605535773811198765209, −0.902560325118226778211369271202, 0.902560325118226778211369271202, 2.36557004605535773811198765209, 3.61601797549595869986265934658, 5.42182873989330627685339758405, 6.08387929336352569408629941974, 7.41630612774521768145736870279, 8.071979864346530587744308799340, 9.422397525258518606352803682510, 9.777227375654841080951682406845, 10.76964383637475038497786200999

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