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Properties

Label 2-378-63.20-c1-0-5
Degree $2$
Conductor $378$
Sign $0.392 - 0.919i$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (1.94 + 3.36i)5-s + (2.09 − 1.60i)7-s + 0.999i·8-s + 3.89i·10-s + (−3.41 − 1.97i)11-s + (−2.46 + 1.42i)13-s + (2.62 − 0.343i)14-s + (−0.5 + 0.866i)16-s + 0.742·17-s − 1.78i·19-s + (−1.94 + 3.36i)20-s + (−1.97 − 3.41i)22-s + (5.41 − 3.12i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.870 + 1.50i)5-s + (0.793 − 0.608i)7-s + 0.353i·8-s + 1.23i·10-s + (−1.03 − 0.594i)11-s + (−0.684 + 0.395i)13-s + (0.701 − 0.0919i)14-s + (−0.125 + 0.216i)16-s + 0.179·17-s − 0.409i·19-s + (−0.435 + 0.753i)20-s + (−0.420 − 0.728i)22-s + (1.12 − 0.651i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.392 - 0.919i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.392 - 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80462 + 1.19259i\)
\(L(\frac12)\) \(\approx\) \(1.80462 + 1.19259i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-2.09 + 1.60i)T \)
good5 \( 1 + (-1.94 - 3.36i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.41 + 1.97i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.46 - 1.42i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.742T + 17T^{2} \)
19 \( 1 + 1.78iT - 19T^{2} \)
23 \( 1 + (-5.41 + 3.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.50 - 1.44i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.04 - 1.75i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.00T + 37T^{2} \)
41 \( 1 + (5.24 + 9.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.471 + 0.816i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.09 + 1.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-0.0105 - 0.0183i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.13 - 1.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.72 + 11.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.94iT - 71T^{2} \)
73 \( 1 - 4.85iT - 73T^{2} \)
79 \( 1 + (1.81 - 3.14i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.02 - 6.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.26T + 89T^{2} \)
97 \( 1 + (16.2 + 9.40i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.26144868622667449547340373046, −10.77446027448526927197054863248, −10.01229365723007970911064529841, −8.607544848163583166647813031779, −7.35466347746792476868784232526, −6.90753060584889673204422505154, −5.72183115976295030765402278971, −4.80596943416824326687101508118, −3.26170456794173020487760776571, −2.27726906169723886926226184739, 1.46945971028880029227670026913, 2.62631417185724535138000997670, 4.61981961141611085277305970270, 5.14981541485944442005752997840, 5.85822237630093570191734312152, 7.55858109744833825527441284287, 8.503376647617699923128331876207, 9.502441909431485061153093772389, 10.19374051481165300861058679345, 11.41716715531014159784390201247

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