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Properties

Label 2-378-189.88-c1-0-11
Degree $2$
Conductor $378$
Sign $0.918 + 0.394i$
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  + (−0.766 − 0.642i)2-s + (1.55 − 0.763i)3-s + (0.173 + 0.984i)4-s + (−1.90 + 1.60i)5-s + (−1.68 − 0.414i)6-s + (2.40 + 1.11i)7-s + (0.500 − 0.866i)8-s + (1.83 − 2.37i)9-s + 2.49·10-s + (−1.91 − 1.60i)11-s + (1.02 + 1.39i)12-s + (2.86 + 1.04i)13-s + (−1.12 − 2.39i)14-s + (−1.74 + 3.94i)15-s + (−0.939 + 0.342i)16-s + 5.20·17-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (0.897 − 0.441i)3-s + (0.0868 + 0.492i)4-s + (−0.853 + 0.716i)5-s + (−0.686 − 0.169i)6-s + (0.907 + 0.419i)7-s + (0.176 − 0.306i)8-s + (0.610 − 0.791i)9-s + 0.787·10-s + (−0.577 − 0.484i)11-s + (0.295 + 0.403i)12-s + (0.794 + 0.289i)13-s + (−0.300 − 0.639i)14-s + (−0.450 + 1.01i)15-s + (−0.234 + 0.0855i)16-s + 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.918 + 0.394i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.918 + 0.394i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35660 - 0.279258i\)
\(L(\frac12)\) \(\approx\) \(1.35660 - 0.279258i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
3 \( 1 + (-1.55 + 0.763i)T \)
7 \( 1 + (-2.40 - 1.11i)T \)
good5 \( 1 + (1.90 - 1.60i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (1.91 + 1.60i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (-2.86 - 1.04i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 - 5.20T + 17T^{2} \)
19 \( 1 - 2.99T + 19T^{2} \)
23 \( 1 + (-8.09 - 2.94i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (7.96 - 2.89i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (1.02 + 5.83i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (2.04 - 3.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (8.59 + 3.12i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.713 + 4.04i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.330 + 1.87i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-3.02 + 5.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (8.05 + 2.93i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.21 - 6.89i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (9.81 - 8.23i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-5.89 - 10.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.81 + 10.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.78 + 3.17i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (2.67 - 0.975i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 4.61T + 89T^{2} \)
97 \( 1 + (-0.823 + 4.67i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.38597015678332036945646596400, −10.49849212001454898099577529635, −9.263980176188333486464601334536, −8.496860881094931312503337070896, −7.65915192010007012696276157996, −7.15961729269322449386009311664, −5.46487503722209289758169524972, −3.69578192372006961564481006084, −3.01087937649878167907306478805, −1.46947212552521719446612254679, 1.35393908510592412597828627607, 3.30972766499857029826722313503, 4.56896422432265880142063025376, 5.33236847630095881917472256229, 7.28282171292137745612234144126, 7.83709615823553501232473146105, 8.504303870958850729190110407572, 9.341479817589400032442448339133, 10.42934996647644182772167042738, 11.11796998838065421802640753451

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