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Properties

Label 2-378-63.32-c2-0-2
Degree $2$
Conductor $378$
Sign $-0.301 - 0.953i$
Analytic cond. $10.2997$
Root an. cond. $3.20932$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s − 3.35i·5-s + (−6.40 + 2.82i)7-s + 2.82i·8-s + (2.37 − 4.10i)10-s + 14.1i·11-s + (−8.31 + 14.4i)13-s + (−9.84 − 1.06i)14-s + (−2.00 + 3.46i)16-s + (13.5 + 7.82i)17-s + (17.3 + 29.9i)19-s + (5.80 − 3.35i)20-s + (−10.0 + 17.3i)22-s − 26.2i·23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s − 0.670i·5-s + (−0.914 + 0.403i)7-s + 0.353i·8-s + (0.237 − 0.410i)10-s + 1.28i·11-s + (−0.639 + 1.10i)13-s + (−0.702 − 0.0761i)14-s + (−0.125 + 0.216i)16-s + (0.797 + 0.460i)17-s + (0.911 + 1.57i)19-s + (0.290 − 0.167i)20-s + (−0.456 + 0.789i)22-s − 1.14i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.301 - 0.953i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.301 - 0.953i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ -0.301 - 0.953i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.07382 + 1.46514i\)
\(L(\frac12)\) \(\approx\) \(1.07382 + 1.46514i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 \)
7 \( 1 + (6.40 - 2.82i)T \)
good5 \( 1 + 3.35iT - 25T^{2} \)
11 \( 1 - 14.1iT - 121T^{2} \)
13 \( 1 + (8.31 - 14.4i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-13.5 - 7.82i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-17.3 - 29.9i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + 26.2iT - 529T^{2} \)
29 \( 1 + (18.2 - 10.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-16.8 - 29.2i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (31.9 + 55.3i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (5.44 + 3.14i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-0.0742 - 0.128i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (36.3 + 20.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (38.6 + 22.2i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (35.9 - 20.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (7.99 - 13.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-42.6 - 73.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 56.4iT - 5.04e3T^{2} \)
73 \( 1 + (-4.42 + 7.67i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (14.5 - 25.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-48.8 + 28.2i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (5.43 - 3.13i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (32.1 + 55.7i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.98592709076704622798815103506, −10.33827273875438985812001773443, −9.576785303897005726080408363756, −8.660285928077930110085387468680, −7.45803781045167882870241538763, −6.63969683874768754044241836379, −5.52264662898139675534267470335, −4.59862182049358999506134888962, −3.45342714732729805408481238460, −1.89310976574514799159834731490, 0.65729529174264273391105678400, 3.06718078323221511549305500011, 3.20598983618699263174575306870, 4.99772028517021062499998721310, 5.95561321259948034504260126704, 6.95495194343244536118455797185, 7.85141818044780619528655086558, 9.393648652684034818335813131814, 10.05552218142418110561623199633, 11.02841105643544483984777648548

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