Properties

Label 2-378-27.4-c1-0-6
Degree $2$
Conductor $378$
Sign $0.711 - 0.703i$
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  + (−0.939 − 0.342i)2-s + (1.71 − 0.214i)3-s + (0.766 + 0.642i)4-s + (−0.701 + 3.97i)5-s + (−1.68 − 0.386i)6-s + (0.766 − 0.642i)7-s + (−0.500 − 0.866i)8-s + (2.90 − 0.737i)9-s + (2.01 − 3.49i)10-s + (−0.324 − 1.84i)11-s + (1.45 + 0.940i)12-s + (0.134 − 0.0489i)13-s + (−0.939 + 0.342i)14-s + (−0.352 + 6.98i)15-s + (0.173 + 0.984i)16-s + (−2.90 + 5.03i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (0.992 − 0.123i)3-s + (0.383 + 0.321i)4-s + (−0.313 + 1.77i)5-s + (−0.689 − 0.157i)6-s + (0.289 − 0.242i)7-s + (−0.176 − 0.306i)8-s + (0.969 − 0.245i)9-s + (0.638 − 1.10i)10-s + (−0.0979 − 0.555i)11-s + (0.419 + 0.271i)12-s + (0.0373 − 0.0135i)13-s + (−0.251 + 0.0914i)14-s + (−0.0909 + 1.80i)15-s + (0.0434 + 0.246i)16-s + (−0.704 + 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.711 - 0.703i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.711 - 0.703i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24763 + 0.512574i\)
\(L(\frac12)\) \(\approx\) \(1.24763 + 0.512574i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 0.342i)T \)
3 \( 1 + (-1.71 + 0.214i)T \)
7 \( 1 + (-0.766 + 0.642i)T \)
good5 \( 1 + (0.701 - 3.97i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (0.324 + 1.84i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-0.134 + 0.0489i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (2.90 - 5.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.89 - 5.02i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.84 - 3.22i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-0.917 - 0.333i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-0.339 - 0.284i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (-5.61 + 9.71i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.19 - 1.89i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.36 + 7.73i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-0.648 + 0.544i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 - 1.33T + 53T^{2} \)
59 \( 1 + (-1.11 + 6.29i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (3.40 - 2.85i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (10.4 - 3.81i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-8.19 + 14.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.37 + 4.10i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-13.0 - 4.76i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (5.71 + 2.08i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (7.33 + 12.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.20 - 6.85i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.08931832809781877083416992453, −10.58666797138606984708882840326, −9.752873233163737997298741428610, −8.607061134731022219856896073135, −7.75027657079626296804100141789, −7.13116845508673246584666119700, −6.11324794771159390392118762527, −3.85129595686895411304415708020, −3.17072233000105987731955856039, −1.93202047470728197306742524262, 1.12287684533827572201688538925, 2.65760509903550317201465595517, 4.56137345841691263352278168078, 5.01839005419895750567134193378, 6.87847733633078719186843742317, 7.84436737148843373296736110850, 8.611510854306325328843640998056, 9.203404584869898153685057284179, 9.806328963766512156251666754824, 11.27937778392078358875127359315

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