Properties

Label 2-378-189.67-c1-0-19
Degree $2$
Conductor $378$
Sign $-0.231 + 0.972i$
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  + (0.766 − 0.642i)2-s + (0.654 − 1.60i)3-s + (0.173 − 0.984i)4-s + (0.190 − 1.08i)5-s + (−0.529 − 1.64i)6-s + (2.53 + 0.751i)7-s + (−0.500 − 0.866i)8-s + (−2.14 − 2.09i)9-s + (−0.549 − 0.951i)10-s + (0.0294 + 0.166i)11-s + (−1.46 − 0.922i)12-s + (−0.287 + 1.63i)13-s + (2.42 − 1.05i)14-s + (−1.61 − 1.01i)15-s + (−0.939 − 0.342i)16-s + (0.236 + 0.409i)17-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (0.377 − 0.925i)3-s + (0.0868 − 0.492i)4-s + (0.0853 − 0.484i)5-s + (−0.216 − 0.673i)6-s + (0.958 + 0.284i)7-s + (−0.176 − 0.306i)8-s + (−0.714 − 0.699i)9-s + (−0.173 − 0.301i)10-s + (0.00887 + 0.0503i)11-s + (−0.423 − 0.266i)12-s + (−0.0797 + 0.452i)13-s + (0.648 − 0.281i)14-s + (−0.415 − 0.261i)15-s + (−0.234 − 0.0855i)16-s + (0.0573 + 0.0992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.231 + 0.972i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.231 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31139 - 1.66061i\)
\(L(\frac12)\) \(\approx\) \(1.31139 - 1.66061i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
3 \( 1 + (-0.654 + 1.60i)T \)
7 \( 1 + (-2.53 - 0.751i)T \)
good5 \( 1 + (-0.190 + 1.08i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-0.0294 - 0.166i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (0.287 - 1.63i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.236 - 0.409i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.72 - 2.98i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.39 + 2.00i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (0.0779 + 0.442i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (0.131 - 0.745i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + 3.02T + 37T^{2} \)
41 \( 1 + (0.223 - 1.26i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-4.12 + 3.46i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-0.836 - 4.74i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-1.65 + 2.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.62 + 1.31i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-2.36 - 13.3i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-10.2 - 8.63i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-6.29 + 10.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 + (-1.86 + 1.56i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (2.32 + 13.1i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (0.437 - 0.757i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.50 + 2.94i)T + (16.8 - 95.5i)T^{2} \)
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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

−11.43554581600339030738998880918, −10.36716737603892564163765029364, −9.060243494433891831909094736159, −8.388821802812574692912956833151, −7.34975959791019726029074625178, −6.19339221905317136740526466137, −5.19110400358596538177870791462, −4.00128524869815444599149182086, −2.44685643713525150111463404739, −1.38967009288812998420548524392, 2.47790961670094481977251814800, 3.72054977239115800660680733252, 4.73417004050267003611386373403, 5.57158141576988455992960336721, 6.91258413735738068763995540283, 7.947847429894828134730674006675, 8.695598205434279905282154289414, 9.868436163187087616385720345877, 10.82815193837165380343519807983, 11.37637738496304269178165502878

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