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Properties

Label 2-378-7.2-c1-0-3
Degree $2$
Conductor $378$
Sign $0.605 - 0.795i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2 − 1.73i)7-s + 0.999·8-s + (3 + 5.19i)11-s + 5·13-s + (2.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (3 + 5.19i)17-s + (2 − 3.46i)19-s − 6·22-s + (3 − 5.19i)23-s + (2.5 + 4.33i)25-s + (−2.5 + 4.33i)26-s + (−0.5 + 2.59i)28-s − 6·29-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.755 − 0.654i)7-s + 0.353·8-s + (0.904 + 1.56i)11-s + 1.38·13-s + (0.668 − 0.231i)14-s + (−0.125 + 0.216i)16-s + (0.727 + 1.26i)17-s + (0.458 − 0.794i)19-s − 1.27·22-s + (0.625 − 1.08i)23-s + (0.5 + 0.866i)25-s + (−0.490 + 0.849i)26-s + (−0.0944 + 0.490i)28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01382 + 0.502581i\)
\(L(\frac12)\) \(\approx\) \(1.01382 + 0.502581i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (2 + 1.73i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 17T + 97T^{2} \)
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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

−11.27631582234505035165473469028, −10.43085265417937114424878775208, −9.531046192159425580107892347213, −8.839353728286661673223844334833, −7.58408794991172284263603141879, −6.81021990656009385284461236775, −6.03218644726813161367450745043, −4.56963000854283076005178933011, −3.53580756699909468184437168160, −1.37362034208339878969768183020, 1.09166412561219319368874002181, 3.04911296347499111336499601451, 3.70065091357188980845042566739, 5.55077881586003523675413321282, 6.30972283644879131055112701675, 7.68325215065703292150497688777, 8.851210752055583988970725354906, 9.222925061899233753146009745160, 10.32809103205095025058943289361, 11.46598164556371743653553111411

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