Properties

Label 2-378-27.16-c1-0-5
Degree $2$
Conductor $378$
Sign $0.622 - 0.782i$
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 + (0.992 + 1.41i)3-s + (0.173 + 0.984i)4-s + (0.999 + 0.363i)5-s + (0.152 − 1.72i)6-s + (−0.173 + 0.984i)7-s + (0.500 − 0.866i)8-s + (−1.03 + 2.81i)9-s + (−0.531 − 0.921i)10-s + (4.14 − 1.50i)11-s + (−1.22 + 1.22i)12-s + (−2.35 + 1.97i)13-s + (0.766 − 0.642i)14-s + (0.475 + 1.78i)15-s + (−0.939 + 0.342i)16-s + (0.333 + 0.578i)17-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (0.572 + 0.819i)3-s + (0.0868 + 0.492i)4-s + (0.447 + 0.162i)5-s + (0.0621 − 0.704i)6-s + (−0.0656 + 0.372i)7-s + (0.176 − 0.306i)8-s + (−0.343 + 0.939i)9-s + (−0.168 − 0.291i)10-s + (1.24 − 0.454i)11-s + (−0.353 + 0.353i)12-s + (−0.652 + 0.547i)13-s + (0.204 − 0.171i)14-s + (0.122 + 0.459i)15-s + (−0.234 + 0.0855i)16-s + (0.0809 + 0.140i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.622 - 0.782i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.622 - 0.782i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19965 + 0.579105i\)
\(L(\frac12)\) \(\approx\) \(1.19965 + 0.579105i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.992 - 1.41i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
good5 \( 1 + (-0.999 - 0.363i)T + (3.83 + 3.21i)T^{2} \)
11 \( 1 + (-4.14 + 1.50i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (2.35 - 1.97i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.333 - 0.578i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.367 + 0.637i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.18 - 6.73i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-1.43 - 1.20i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.0404 + 0.229i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (-3.60 - 6.23i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.28 + 6.95i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (7.07 - 2.57i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-1.85 + 10.5i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 + (7.20 + 2.62i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.0669 + 0.379i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-5.54 + 4.65i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.205 - 0.355i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.24 + 9.07i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.60 + 7.21i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (9.01 + 7.56i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (0.612 - 1.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.0 - 5.12i)T + (74.3 - 62.3i)T^{2} \)
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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

−11.44238566026926547160821088845, −10.35939273529317271645955806289, −9.531513359134655701454717397999, −9.094186720611141465312649254595, −8.110490778637511514637556069190, −6.89624508564842379227055771840, −5.61885975297480890558703521814, −4.25109253051187980106023021469, −3.19514922540507771440334930714, −1.92019953678949994193098644438, 1.11426541522230063067801729219, 2.53648597551416542984628681990, 4.21892941842330540272959295355, 5.78057054792255926126464675400, 6.71592949044323375709232541164, 7.45068772738126612981702199985, 8.397349950357341371599555349589, 9.349200649359226397204232495835, 9.915290945990282275816981177626, 11.21810675608218680373791521519

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