Properties

Label 2-378-63.5-c1-0-3
Degree $2$
Conductor $378$
Sign $0.988 + 0.152i$
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 + (0.450 + 0.779i)5-s + (1.57 + 2.12i)7-s + i·8-s + (0.779 − 0.450i)10-s + (2.70 + 1.56i)11-s + (−1.99 − 1.14i)13-s + (2.12 − 1.57i)14-s + 16-s + (2.57 + 4.46i)17-s + (2.38 + 1.37i)19-s + (−0.450 − 0.779i)20-s + (1.56 − 2.70i)22-s + (1.48 − 0.857i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.201 + 0.348i)5-s + (0.593 + 0.804i)7-s + 0.353i·8-s + (0.246 − 0.142i)10-s + (0.816 + 0.471i)11-s + (−0.552 − 0.318i)13-s + (0.568 − 0.420i)14-s + 0.250·16-s + (0.624 + 1.08i)17-s + (0.546 + 0.315i)19-s + (−0.100 − 0.174i)20-s + (0.333 − 0.577i)22-s + (0.309 − 0.178i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.988 + 0.152i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.988 + 0.152i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43513 - 0.110096i\)
\(L(\frac12)\) \(\approx\) \(1.43513 - 0.110096i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.57 - 2.12i)T \)
good5 \( 1 + (-0.450 - 0.779i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.70 - 1.56i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.99 + 1.14i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.57 - 4.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.38 - 1.37i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.48 + 0.857i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.85 + 1.07i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.0iT - 31T^{2} \)
37 \( 1 + (4.73 - 8.20i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.22 - 2.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.273 + 0.473i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.86T + 47T^{2} \)
53 \( 1 + (-12.0 + 6.97i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 7.98T + 59T^{2} \)
61 \( 1 - 7.25iT - 61T^{2} \)
67 \( 1 - 3.67T + 67T^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 + (10.9 - 6.30i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 6.54T + 79T^{2} \)
83 \( 1 + (0.184 + 0.319i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.00 - 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.86 + 5.12i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.54343325525084148957313394860, −10.33907068368139829864662155745, −9.752643521869336944092291273326, −8.663976360742306434984380906547, −7.83098794940460600755861171129, −6.45360351820606270942345888363, −5.38920235050415885503327087294, −4.25523207858031335932756944130, −2.88978995156385301981488344678, −1.65542949934827996891396435767, 1.18060656934379677868404575002, 3.39260258054108577473071266735, 4.71893551264702974344415610573, 5.43742496286915665937960720466, 6.91632178222776114024197762158, 7.38323053197321262456443825251, 8.660019981016677605730426324385, 9.320120395175277754143979144586, 10.37058507166914367682938138914, 11.43051253889460082554062950879

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