Properties

Label 2-378-189.41-c1-0-9
Degree $2$
Conductor $378$
Sign $0.581 - 0.813i$
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  + (−0.984 + 0.173i)2-s + (1.56 + 0.743i)3-s + (0.939 − 0.342i)4-s + (0.941 + 0.789i)5-s + (−1.66 − 0.460i)6-s + (2.22 + 1.42i)7-s + (−0.866 + 0.5i)8-s + (1.89 + 2.32i)9-s + (−1.06 − 0.614i)10-s + (−1.50 − 1.79i)11-s + (1.72 + 0.163i)12-s + (−0.429 − 0.0757i)13-s + (−2.44 − 1.01i)14-s + (0.885 + 1.93i)15-s + (0.766 − 0.642i)16-s + (−1.74 + 3.01i)17-s + ⋯
L(s)  = 1  + (−0.696 + 0.122i)2-s + (0.903 + 0.429i)3-s + (0.469 − 0.171i)4-s + (0.420 + 0.353i)5-s + (−0.681 − 0.187i)6-s + (0.842 + 0.539i)7-s + (−0.306 + 0.176i)8-s + (0.631 + 0.775i)9-s + (−0.336 − 0.194i)10-s + (−0.454 − 0.541i)11-s + (0.497 + 0.0471i)12-s + (−0.119 − 0.0210i)13-s + (−0.652 − 0.272i)14-s + (0.228 + 0.499i)15-s + (0.191 − 0.160i)16-s + (−0.422 + 0.732i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.581 - 0.813i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.581 - 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33184 + 0.685332i\)
\(L(\frac12)\) \(\approx\) \(1.33184 + 0.685332i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 - 0.173i)T \)
3 \( 1 + (-1.56 - 0.743i)T \)
7 \( 1 + (-2.22 - 1.42i)T \)
good5 \( 1 + (-0.941 - 0.789i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (1.50 + 1.79i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.429 + 0.0757i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.74 - 3.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.97 + 1.13i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.85 + 5.08i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (2.08 - 0.367i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.87 - 7.89i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (-2.71 + 4.69i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.273 - 1.55i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (1.56 - 1.31i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (4.45 + 1.62i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 0.117iT - 53T^{2} \)
59 \( 1 + (-9.22 - 7.74i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-3.82 + 10.5i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.46 + 8.32i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (1.77 + 1.02i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.00 - 3.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.916 + 5.19i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (1.50 + 8.54i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (7.35 + 12.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.82 + 11.7i)T + (-16.8 + 95.5i)T^{2} \)
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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

−11.09414252557533280907737082591, −10.46997360413255804778067512228, −9.618362451397109730595049608882, −8.508815357915959237201792174926, −8.249840294386934360962508987724, −7.00357462997559988668122211852, −5.77199078763602609990770953565, −4.56405519590258237518141971828, −2.94912879826454823726629267140, −1.93596447447321582831915118092, 1.35521641805480527688906576340, 2.46812369593710793984982507098, 4.01680667305921378942533217941, 5.38907731966632255271727047039, 6.93883025330599919386080170295, 7.67776783194550084110978565418, 8.346628844551710538569374380715, 9.527072474041860157856731539452, 9.881908607545885509464566932793, 11.23783460097241701449355986259

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