Properties

Label 2-378-27.7-c1-0-8
Degree $2$
Conductor $378$
Sign $0.861 + 0.507i$
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 + (0.927 − 1.46i)3-s + (0.766 − 0.642i)4-s + (0.587 + 3.33i)5-s + (0.371 − 1.69i)6-s + (0.766 + 0.642i)7-s + (0.500 − 0.866i)8-s + (−1.27 − 2.71i)9-s + (1.69 + 2.93i)10-s + (0.344 − 1.95i)11-s + (−0.229 − 1.71i)12-s + (4.04 + 1.47i)13-s + (0.939 + 0.342i)14-s + (5.41 + 2.23i)15-s + (0.173 − 0.984i)16-s + (−2.73 − 4.72i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (0.535 − 0.844i)3-s + (0.383 − 0.321i)4-s + (0.262 + 1.49i)5-s + (0.151 − 0.690i)6-s + (0.289 + 0.242i)7-s + (0.176 − 0.306i)8-s + (−0.426 − 0.904i)9-s + (0.535 + 0.926i)10-s + (0.103 − 0.588i)11-s + (−0.0662 − 0.495i)12-s + (1.12 + 0.408i)13-s + (0.251 + 0.0914i)14-s + (1.39 + 0.576i)15-s + (0.0434 − 0.246i)16-s + (−0.662 − 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.861 + 0.507i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.861 + 0.507i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.34430 - 0.638897i\)
\(L(\frac12)\) \(\approx\) \(2.34430 - 0.638897i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.927 + 1.46i)T \)
7 \( 1 + (-0.766 - 0.642i)T \)
good5 \( 1 + (-0.587 - 3.33i)T + (-4.69 + 1.71i)T^{2} \)
11 \( 1 + (-0.344 + 1.95i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-4.04 - 1.47i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (2.73 + 4.72i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.16 - 5.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.24 + 1.04i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (8.55 - 3.11i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-5.08 + 4.26i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (1.38 + 2.39i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.31 + 2.66i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.0470 - 0.266i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-4.41 - 3.70i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 + (-1.81 - 10.2i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-1.50 - 1.26i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-0.410 - 0.149i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-5.21 - 9.03i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.40 + 4.15i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.80 - 1.38i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-12.2 + 4.45i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-3.08 + 5.33i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.77 - 10.0i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.28209738921821711370227652961, −10.78007363826322975051011444924, −9.463769471198313433126154906010, −8.400831841543029680261457638134, −7.26893450366757119165105777591, −6.47230560711395932420553072418, −5.79482246922809120340438853984, −3.87577293784580753085702883130, −2.92966142532336346515363422814, −1.86931871058576259815598243587, 1.86959613155656120716895567628, 3.66351745672087832072678488194, 4.57205768034222525917488991328, 5.21563127045263187627961828490, 6.45516605773027202012165682917, 8.036141986807895170020809426302, 8.622464142024932096783578129145, 9.416070253569248346261417840164, 10.60679105136642681335178635384, 11.38255623057027882883928600457

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