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Properties

Label 2-378-63.4-c1-0-3
Degree $2$
Conductor $378$
Sign $0.609 + 0.792i$
Analytic cond. $3.01834$
Root an. cond. $1.73733$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + 0.460·5-s + (2.25 + 1.38i)7-s − 0.999·8-s + (0.230 − 0.398i)10-s + 3.64·11-s + (0.730 − 1.26i)13-s + (2.32 − 1.26i)14-s + (−0.5 + 0.866i)16-s + (1.86 − 3.23i)17-s + (−2.02 − 3.51i)19-s + (−0.230 − 0.398i)20-s + (1.82 − 3.15i)22-s − 1.13·23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.205·5-s + (0.853 + 0.521i)7-s − 0.353·8-s + (0.0728 − 0.126i)10-s + 1.09·11-s + (0.202 − 0.350i)13-s + (0.621 − 0.338i)14-s + (−0.125 + 0.216i)16-s + (0.452 − 0.784i)17-s + (−0.465 − 0.805i)19-s + (−0.0514 − 0.0891i)20-s + (0.388 − 0.673i)22-s − 0.236·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.609 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.609 + 0.792i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.609 + 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62452 - 0.800440i\)
\(L(\frac12)\) \(\approx\) \(1.62452 - 0.800440i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-2.25 - 1.38i)T \)
good5 \( 1 - 0.460T + 5T^{2} \)
11 \( 1 - 3.64T + 11T^{2} \)
13 \( 1 + (-0.730 + 1.26i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.86 + 3.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.02 + 3.51i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.13T + 23T^{2} \)
29 \( 1 + (-4.48 - 7.77i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.257 - 0.445i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.55 + 7.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.472 + 0.819i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.66 - 8.07i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.16 + 2.01i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.21 - 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.44 + 11.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.04 - 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.16 - 2.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.67T + 71T^{2} \)
73 \( 1 + (6.62 - 11.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.50 + 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.32 + 5.75i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.36 - 2.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.59 + 9.68i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.32686969203929651868753031976, −10.54093751102804657654714834640, −9.349615211432034329660144714393, −8.755671667260364153794665974574, −7.48967446443218884831093625773, −6.22491978968691802364842332566, −5.22398128177574661142213873457, −4.21794801333404956715514154573, −2.83295473732128217190212672836, −1.44616186125906097032326609181, 1.69182270299532967628053410314, 3.74481133506278839689758415473, 4.50045599671527860788932802720, 5.85594484002507451364554872461, 6.60231345405632272593502687381, 7.81403251583164097095617833192, 8.444779936390306220077431060094, 9.623116016839618038712585132029, 10.57048824813704249887219649070, 11.71298199121324714001234439539

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