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Properties

Label 2-378-189.101-c1-0-10
Degree $2$
Conductor $378$
Sign $0.985 - 0.169i$
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  + (0.642 − 0.766i)2-s + (1.51 + 0.835i)3-s + (−0.173 − 0.984i)4-s + (−1.65 + 1.39i)5-s + (1.61 − 0.625i)6-s + (1.78 + 1.95i)7-s + (−0.866 − 0.500i)8-s + (1.60 + 2.53i)9-s + 2.16i·10-s + (3.26 − 3.89i)11-s + (0.559 − 1.63i)12-s + (−1.92 + 5.27i)13-s + (2.64 − 0.108i)14-s + (−3.67 + 0.726i)15-s + (−0.939 + 0.342i)16-s + 3.40·17-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (0.876 + 0.482i)3-s + (−0.0868 − 0.492i)4-s + (−0.741 + 0.622i)5-s + (0.659 − 0.255i)6-s + (0.673 + 0.738i)7-s + (−0.306 − 0.176i)8-s + (0.534 + 0.844i)9-s + 0.684i·10-s + (0.984 − 1.17i)11-s + (0.161 − 0.473i)12-s + (−0.532 + 1.46i)13-s + (0.706 − 0.0290i)14-s + (−0.950 + 0.187i)15-s + (−0.234 + 0.0855i)16-s + 0.826·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.985 - 0.169i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.985 - 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13363 + 0.182490i\)
\(L(\frac12)\) \(\approx\) \(2.13363 + 0.182490i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 + 0.766i)T \)
3 \( 1 + (-1.51 - 0.835i)T \)
7 \( 1 + (-1.78 - 1.95i)T \)
good5 \( 1 + (1.65 - 1.39i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-3.26 + 3.89i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (1.92 - 5.27i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 3.40T + 17T^{2} \)
19 \( 1 + 5.51iT - 19T^{2} \)
23 \( 1 + (-0.908 + 2.49i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.75 + 4.82i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (8.84 - 1.55i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-2.24 + 3.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.62 + 2.04i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.146 - 0.829i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.707 + 4.01i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (4.18 + 2.41i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.18 + 0.432i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.81 + 0.320i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-9.46 + 7.94i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.143 - 0.0829i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.45 - 1.99i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.36 + 5.34i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-12.0 + 4.37i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 6.94T + 89T^{2} \)
97 \( 1 + (-14.0 - 2.46i)T + (91.1 + 33.1i)T^{2} \)
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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

−11.41749064449260547104996661094, −10.77507825997791873301079500138, −9.316865922562213325700884649168, −8.952153370739043524307633046728, −7.75941009711126489161083524471, −6.67753372965561300867160312013, −5.20838652887421064983920111948, −4.10561855535421820176780047235, −3.26314990576538074041885874434, −2.03393772774429997382044194539, 1.45526347699617219966170168512, 3.43632943387916215844834548082, 4.23883791593150447235318262277, 5.37780620281370726743079872326, 6.93242758312902362689628496906, 7.76396523454216182127234511262, 8.049519359289583062088697385147, 9.304203438449020132244067173301, 10.26911265304521187228283556227, 11.77313976885216206722460231119

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