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Properties

Label 2-378-189.101-c1-0-11
Degree $2$
Conductor $378$
Sign $0.558 + 0.829i$
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 + (−1.58 − 0.707i)3-s + (−0.173 − 0.984i)4-s + (2.18 − 1.83i)5-s + (1.55 − 0.756i)6-s + (−1.61 + 2.09i)7-s + (0.866 + 0.500i)8-s + (2.00 + 2.23i)9-s + 2.85i·10-s + (−0.926 + 1.10i)11-s + (−0.421 + 1.67i)12-s + (2.10 − 5.77i)13-s + (−0.563 − 2.58i)14-s + (−4.75 + 1.35i)15-s + (−0.939 + 0.342i)16-s − 0.818·17-s + ⋯
L(s)  = 1  + (−0.454 + 0.541i)2-s + (−0.912 − 0.408i)3-s + (−0.0868 − 0.492i)4-s + (0.978 − 0.821i)5-s + (0.636 − 0.308i)6-s + (−0.611 + 0.791i)7-s + (0.306 + 0.176i)8-s + (0.666 + 0.745i)9-s + 0.903i·10-s + (−0.279 + 0.332i)11-s + (−0.121 + 0.484i)12-s + (0.582 − 1.60i)13-s + (−0.150 − 0.690i)14-s + (−1.22 + 0.350i)15-s + (−0.234 + 0.0855i)16-s − 0.198·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.558 + 0.829i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.558 + 0.829i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.708165 - 0.376877i\)
\(L(\frac12)\) \(\approx\) \(0.708165 - 0.376877i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 + 0.707i)T \)
7 \( 1 + (1.61 - 2.09i)T \)
good5 \( 1 + (-2.18 + 1.83i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (0.926 - 1.10i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-2.10 + 5.77i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 0.818T + 17T^{2} \)
19 \( 1 + 7.47iT - 19T^{2} \)
23 \( 1 + (-1.97 + 5.43i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.98 - 5.44i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (1.41 - 0.250i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-4.32 + 7.49i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.74 - 0.998i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.88 + 10.6i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.657 - 3.72i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (4.45 + 2.56i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.22 - 2.63i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (5.67 + 0.999i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (7.60 - 6.38i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (12.4 - 7.21i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.448 + 0.258i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-11.2 - 9.41i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-10.9 + 3.97i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 8.45T + 89T^{2} \)
97 \( 1 + (6.43 + 1.13i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.97412876651656161682802102614, −10.31392898762163404031705508049, −9.240451651665426960266535004281, −8.600657515832817400110438363297, −7.30341083843350055570516474396, −6.29804223576238380542321163673, −5.54039362551737128285383035407, −4.91249241376524601896294604562, −2.45966256661583137552905415525, −0.74305345452155461952139154749, 1.56620004917470006610510969788, 3.32755637002838393725210940017, 4.37350477089868525311612935167, 6.06059240986919691110349750850, 6.48060883776710393209122609821, 7.68269364074028980700363538827, 9.319695604554761247834131089368, 9.834985527416152835761272556813, 10.56030729439001087440782390039, 11.25334496255279700191052439397

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