Properties

Label 2-378-1.1-c1-0-4
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 + 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: $\chi_{378} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.398893991\)
\(L(\frac12)\) \(\approx\) \(2.398893991\)
\(L(\frac{3}{2})\) 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

−11.42061259594256530369121590340, −10.26708504332289108173004023441, −9.938898618957151166860023695517, −8.539023583449884225816408011720, −7.46960982382182631181281303264, −6.34863993594311576079450533659, −5.42330356244252482781728099229, −4.70886712751954702153319801692, −2.95503675125733963986543239116, −1.90035837414507085552956582982, 1.90035837414507085552956582982, 2.95503675125733963986543239116, 4.70886712751954702153319801692, 5.42330356244252482781728099229, 6.34863993594311576079450533659, 7.46960982382182631181281303264, 8.539023583449884225816408011720, 9.938898618957151166860023695517, 10.26708504332289108173004023441, 11.42061259594256530369121590340

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