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Properties

Label 2-378-21.20-c1-0-9
Degree $2$
Conductor $378$
Sign $-0.327 + 0.944i$
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  − i·2-s − 4-s + 1.73·5-s + (−2.5 − 0.866i)7-s + i·8-s − 1.73i·10-s − 3i·11-s − 6.92i·13-s + (−0.866 + 2.5i)14-s + 16-s + 6.92·17-s − 3.46i·19-s − 1.73·20-s − 3·22-s + 6i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.774·5-s + (−0.944 − 0.327i)7-s + 0.353i·8-s − 0.547i·10-s − 0.904i·11-s − 1.92i·13-s + (−0.231 + 0.668i)14-s + 0.250·16-s + 1.68·17-s − 0.794i·19-s − 0.387·20-s − 0.639·22-s + 1.25i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.327 + 0.944i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.327 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.733330 - 1.03011i\)
\(L(\frac12)\) \(\approx\) \(0.733330 - 1.03011i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 + (2.5 + 0.866i)T \)
good5 \( 1 - 1.73T + 5T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
13 \( 1 + 6.92iT - 13T^{2} \)
17 \( 1 - 6.92T + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 5.19iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 3iT - 53T^{2} \)
59 \( 1 - 3.46T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 - 12.1iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 1.73T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + 12.1iT - 97T^{2} \)
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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

−10.94054348308868652549869382528, −10.03428170579697789839389304342, −9.710099163967345657983701077059, −8.439673463199116733588540497238, −7.44445268402087159160073099768, −5.91496784936962241399101036041, −5.42256276793357544650064973651, −3.55858479146466087117254049237, −2.86694919426937825445757505001, −0.899611126626033953425390149797, 1.97329068957981973631281959760, 3.68526917680665035907942788732, 4.96325070707529899734466334736, 6.10793565913202028806178696743, 6.70024093427145763703808662451, 7.77949847293907118173619831348, 9.090393223844280648885121274127, 9.624812580510742038863350211755, 10.33840165168140928444411217414, 11.95897178575845675128622839395

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