Properties

Label 2-378-1.1-c3-0-14
Degree $2$
Conductor $378$
Sign $-1$
Analytic cond. $22.3027$
Root an. cond. $4.72257$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s − 15.9·5-s + 7·7-s − 8·8-s + 31.8·10-s − 2.05·11-s + 79.6·13-s − 14·14-s + 16·16-s + 67.5·17-s − 108.·19-s − 63.7·20-s + 4.10·22-s + 7.74·23-s + 129.·25-s − 159.·26-s + 28·28-s − 249.·29-s + 5.30·31-s − 32·32-s − 135.·34-s − 111.·35-s − 48.6·37-s + 216.·38-s + 127.·40-s + 45.0·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.42·5-s + 0.377·7-s − 0.353·8-s + 1.00·10-s − 0.0561·11-s + 1.70·13-s − 0.267·14-s + 0.250·16-s + 0.964·17-s − 1.30·19-s − 0.713·20-s + 0.0397·22-s + 0.0702·23-s + 1.03·25-s − 1.20·26-s + 0.188·28-s − 1.59·29-s + 0.0307·31-s − 0.176·32-s − 0.681·34-s − 0.539·35-s − 0.216·37-s + 0.925·38-s + 0.504·40-s + 0.171·41-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 378,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
3 \( 1 \)
7 \( 1 - 7T \)
good5 \( 1 + 15.9T + 125T^{2} \)
11 \( 1 + 2.05T + 1.33e3T^{2} \)
13 \( 1 - 79.6T + 2.19e3T^{2} \)
17 \( 1 - 67.5T + 4.91e3T^{2} \)
19 \( 1 + 108.T + 6.85e3T^{2} \)
23 \( 1 - 7.74T + 1.21e4T^{2} \)
29 \( 1 + 249.T + 2.43e4T^{2} \)
31 \( 1 - 5.30T + 2.97e4T^{2} \)
37 \( 1 + 48.6T + 5.06e4T^{2} \)
41 \( 1 - 45.0T + 6.89e4T^{2} \)
43 \( 1 + 201.T + 7.95e4T^{2} \)
47 \( 1 - 4.49T + 1.03e5T^{2} \)
53 \( 1 + 279.T + 1.48e5T^{2} \)
59 \( 1 + 675.T + 2.05e5T^{2} \)
61 \( 1 + 842.T + 2.26e5T^{2} \)
67 \( 1 - 329.T + 3.00e5T^{2} \)
71 \( 1 - 811.T + 3.57e5T^{2} \)
73 \( 1 - 635.T + 3.89e5T^{2} \)
79 \( 1 + 480.T + 4.93e5T^{2} \)
83 \( 1 + 1.00e3T + 5.71e5T^{2} \)
89 \( 1 + 1.37e3T + 7.04e5T^{2} \)
97 \( 1 + 1.67e3T + 9.12e5T^{2} \)
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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

−10.87548278201405929715018866684, −9.419328707771239713712968757226, −8.339762220389741841391555892341, −8.007317160303903717525172164971, −6.92121125663947098563524768211, −5.77360164986486181078762450499, −4.21785657137222293266457649852, −3.33710530503998290147402861982, −1.47217450382155596059329704638, 0, 1.47217450382155596059329704638, 3.33710530503998290147402861982, 4.21785657137222293266457649852, 5.77360164986486181078762450499, 6.92121125663947098563524768211, 8.007317160303903717525172164971, 8.339762220389741841391555892341, 9.419328707771239713712968757226, 10.87548278201405929715018866684

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