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Properties

Label 2-378-189.131-c1-0-6
Degree $2$
Conductor $378$
Sign $0.234 - 0.972i$
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  + (−0.642 − 0.766i)2-s + (0.0131 + 1.73i)3-s + (−0.173 + 0.984i)4-s + (3.23 + 2.71i)5-s + (1.31 − 1.12i)6-s + (1.66 + 2.05i)7-s + (0.866 − 0.500i)8-s + (−2.99 + 0.0454i)9-s − 4.22i·10-s + (−0.952 − 1.13i)11-s + (−1.70 − 0.287i)12-s + (−0.0656 − 0.180i)13-s + (0.502 − 2.59i)14-s + (−4.66 + 5.64i)15-s + (−0.939 − 0.342i)16-s − 2.02·17-s + ⋯
L(s)  = 1  + (−0.454 − 0.541i)2-s + (0.00757 + 0.999i)3-s + (−0.0868 + 0.492i)4-s + (1.44 + 1.21i)5-s + (0.538 − 0.458i)6-s + (0.629 + 0.776i)7-s + (0.306 − 0.176i)8-s + (−0.999 + 0.0151i)9-s − 1.33i·10-s + (−0.287 − 0.342i)11-s + (−0.493 − 0.0830i)12-s + (−0.0182 − 0.0500i)13-s + (0.134 − 0.694i)14-s + (−1.20 + 1.45i)15-s + (−0.234 − 0.0855i)16-s − 0.489·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.234 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.234 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.234 - 0.972i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.234 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04787 + 0.824860i\)
\(L(\frac12)\) \(\approx\) \(1.04787 + 0.824860i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 + 0.766i)T \)
3 \( 1 + (-0.0131 - 1.73i)T \)
7 \( 1 + (-1.66 - 2.05i)T \)
good5 \( 1 + (-3.23 - 2.71i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (0.952 + 1.13i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.0656 + 0.180i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + 2.02T + 17T^{2} \)
19 \( 1 + 2.77iT - 19T^{2} \)
23 \( 1 + (3.18 + 8.73i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (1.08 - 2.97i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-8.59 - 1.51i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (-0.880 - 1.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.23 + 2.99i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.731 - 4.15i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (1.78 + 10.1i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-0.697 + 0.402i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (12.1 - 4.43i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.731 + 0.128i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (-2.31 - 1.94i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.21 + 0.703i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.52 - 2.61i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.61 + 4.71i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-3.70 - 1.34i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + (11.9 - 2.10i)T + (91.1 - 33.1i)T^{2} \)
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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

−11.04231783299677013683612512013, −10.67213242377808284012134175852, −9.872439649919016612982358000723, −9.065876130521485700173905612582, −8.256372806547474316813431882513, −6.63434763201680805281996764036, −5.75037775653281572894942021831, −4.61985009670757196581204836783, −2.91387462176812737255612213344, −2.30589394308591105590384997829, 1.13638733855306502354979981823, 2.06246989421179891254071363337, 4.57500870911669546142840595792, 5.62711577978484450604769673679, 6.31609258620665450829450270609, 7.62106545196625572069552241017, 8.182875264142237383375038974339, 9.290718043117841596984111948050, 9.930450393340362602679031368945, 11.12572178534615114259972445323

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