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Properties

Label 2-378-189.101-c1-0-17
Degree $2$
Conductor $378$
Sign $0.890 + 0.455i$
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 + (1.49 + 0.873i)3-s + (−0.173 − 0.984i)4-s + (2.37 − 1.98i)5-s + (1.63 − 0.584i)6-s + (0.451 + 2.60i)7-s + (−0.866 − 0.500i)8-s + (1.47 + 2.61i)9-s − 3.09i·10-s + (−2.81 + 3.35i)11-s + (0.600 − 1.62i)12-s + (1.81 − 4.99i)13-s + (2.28 + 1.32i)14-s + (5.28 − 0.904i)15-s + (−0.939 + 0.342i)16-s − 7.67·17-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (0.863 + 0.504i)3-s + (−0.0868 − 0.492i)4-s + (1.06 − 0.889i)5-s + (0.665 − 0.238i)6-s + (0.170 + 0.985i)7-s + (−0.306 − 0.176i)8-s + (0.491 + 0.871i)9-s − 0.978i·10-s + (−0.849 + 1.01i)11-s + (0.173 − 0.468i)12-s + (0.503 − 1.38i)13-s + (0.611 + 0.355i)14-s + (1.36 − 0.233i)15-s + (−0.234 + 0.0855i)16-s − 1.86·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.890 + 0.455i$
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: \((2,\ 378,\ (\ :1/2),\ 0.890 + 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.38106 - 0.574192i\)
\(L(\frac12)\) \(\approx\) \(2.38106 - 0.574192i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.49 - 0.873i)T \)
7 \( 1 + (-0.451 - 2.60i)T \)
good5 \( 1 + (-2.37 + 1.98i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (2.81 - 3.35i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-1.81 + 4.99i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 7.67T + 17T^{2} \)
19 \( 1 - 0.862iT - 19T^{2} \)
23 \( 1 + (-0.773 + 2.12i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (3.14 + 8.64i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-1.66 + 0.293i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (2.35 - 4.07i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.88 - 3.59i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.314 - 1.78i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.462 + 2.62i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (2.90 + 1.67i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.29 + 3.38i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-5.22 - 0.921i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (3.96 - 3.32i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-3.61 + 2.08i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.25 - 0.726i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.35 + 1.13i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (12.6 - 4.61i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 5.72T + 89T^{2} \)
97 \( 1 + (-16.1 - 2.85i)T + (91.1 + 33.1i)T^{2} \)
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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

−11.18012516495175534335239642353, −10.21776814207312724526193511316, −9.531114360671979415066068604878, −8.767888241933584541655398408018, −7.923954844033873303165735913455, −6.10604037323364634881446883271, −5.15486786610991923232702566144, −4.43889985336464565667927908969, −2.69802564558646402010402332048, −1.99296043568065911513211799272, 1.98760062185201997348956690028, 3.18780293149847732137635851606, 4.36545300936125720400872511115, 5.96286414756090527070481274460, 6.79531601299886314340507999368, 7.35122764944690647824742573373, 8.661451147363390799100823020250, 9.312099760556830163538356771995, 10.68929524012861157671772603737, 11.20780415401378000763507104159

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