Properties

Label 2-378-3.2-c4-0-15
Degree $2$
Conductor $378$
Sign $-i$
Analytic cond. $39.0738$
Root an. cond. $6.25090$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·2-s − 8.00·4-s + 28.1i·5-s − 18.5·7-s − 22.6i·8-s − 79.7·10-s − 168. i·11-s + 319.·13-s − 52.3i·14-s + 64.0·16-s − 39.1i·17-s + 457.·19-s − 225. i·20-s + 476.·22-s + 435. i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.500·4-s + 1.12i·5-s − 0.377·7-s − 0.353i·8-s − 0.797·10-s − 1.39i·11-s + 1.88·13-s − 0.267i·14-s + 0.250·16-s − 0.135i·17-s + 1.26·19-s − 0.563i·20-s + 0.984·22-s + 0.822i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-i$
Analytic conductor: \(39.0738\)
Root analytic conductor: \(6.25090\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :2),\ -i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.982679539\)
\(L(\frac12)\) \(\approx\) \(1.982679539\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82iT \)
3 \( 1 \)
7 \( 1 + 18.5T \)
good5 \( 1 - 28.1iT - 625T^{2} \)
11 \( 1 + 168. iT - 1.46e4T^{2} \)
13 \( 1 - 319.T + 2.85e4T^{2} \)
17 \( 1 + 39.1iT - 8.35e4T^{2} \)
19 \( 1 - 457.T + 1.30e5T^{2} \)
23 \( 1 - 435. iT - 2.79e5T^{2} \)
29 \( 1 + 135. iT - 7.07e5T^{2} \)
31 \( 1 + 267.T + 9.23e5T^{2} \)
37 \( 1 - 812.T + 1.87e6T^{2} \)
41 \( 1 + 737. iT - 2.82e6T^{2} \)
43 \( 1 + 2.09e3T + 3.41e6T^{2} \)
47 \( 1 - 2.96e3iT - 4.87e6T^{2} \)
53 \( 1 + 3.19e3iT - 7.89e6T^{2} \)
59 \( 1 - 5.62e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.86e3T + 1.38e7T^{2} \)
67 \( 1 + 2.00e3T + 2.01e7T^{2} \)
71 \( 1 - 4.30e3iT - 2.54e7T^{2} \)
73 \( 1 - 6.31e3T + 2.83e7T^{2} \)
79 \( 1 + 3.70e3T + 3.89e7T^{2} \)
83 \( 1 - 9.03e3iT - 4.74e7T^{2} \)
89 \( 1 + 3.71e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.37e4T + 8.85e7T^{2} \)
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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

−11.06485699551796229417757424753, −10.01968629501746567321510718484, −8.968160541574592945990762191030, −8.112587068965606261063546253471, −7.06620695377778370016465405782, −6.19691577013487179154722082794, −5.55951633906580789875979502021, −3.71591298352670983082474059835, −3.10856963233416816976249377835, −0.978598553003939144593399100929, 0.78281982667495288420728384043, 1.78069874509891056203252195031, 3.40177278654584788309302066770, 4.44300102444103114304360814004, 5.37745397214819855024757445209, 6.63513797375301590171830836386, 7.998465798180781423953243244353, 8.859234922150909805820291736171, 9.581532019279355264587961675426, 10.47309855734118655210746670320

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