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Properties

Label 2-378-189.101-c1-0-22
Degree $2$
Conductor $378$
Sign $0.235 + 0.971i$
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 + (1.71 − 0.207i)3-s + (−0.173 − 0.984i)4-s + (1.27 − 1.06i)5-s + (0.946 − 1.45i)6-s + (−1.28 − 2.31i)7-s + (−0.866 − 0.500i)8-s + (2.91 − 0.714i)9-s − 1.66i·10-s + (−0.966 + 1.15i)11-s + (−0.503 − 1.65i)12-s + (−1.85 + 5.09i)13-s + (−2.59 − 0.503i)14-s + (1.96 − 2.10i)15-s + (−0.939 + 0.342i)16-s − 0.973·17-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (0.992 − 0.120i)3-s + (−0.0868 − 0.492i)4-s + (0.568 − 0.477i)5-s + (0.386 − 0.592i)6-s + (−0.485 − 0.874i)7-s + (−0.306 − 0.176i)8-s + (0.971 − 0.238i)9-s − 0.525i·10-s + (−0.291 + 0.347i)11-s + (−0.145 − 0.478i)12-s + (−0.514 + 1.41i)13-s + (−0.694 − 0.134i)14-s + (0.507 − 0.542i)15-s + (−0.234 + 0.0855i)16-s − 0.235·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.235 + 0.971i$
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: \((2,\ 378,\ (\ :1/2),\ 0.235 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83255 - 1.44124i\)
\(L(\frac12)\) \(\approx\) \(1.83255 - 1.44124i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 + 0.207i)T \)
7 \( 1 + (1.28 + 2.31i)T \)
good5 \( 1 + (-1.27 + 1.06i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (0.966 - 1.15i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (1.85 - 5.09i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 0.973T + 17T^{2} \)
19 \( 1 + 2.22iT - 19T^{2} \)
23 \( 1 + (0.445 - 1.22i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-2.35 - 6.48i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-10.9 + 1.92i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-2.90 + 5.02i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (8.50 + 3.09i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.660 + 3.74i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (1.70 - 9.65i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (3.84 + 2.21i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.65 + 2.42i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-8.70 - 1.53i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-1.82 + 1.52i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-4.03 + 2.32i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.56 - 4.94i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.94 - 1.63i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (14.0 - 5.11i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 5.09T + 89T^{2} \)
97 \( 1 + (13.8 + 2.43i)T + (91.1 + 33.1i)T^{2} \)
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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

−11.17265523860070750426311500522, −9.986726973810184282391344356535, −9.560316064339956971028351875220, −8.643032746742509682266766755463, −7.28684901585302901975617384367, −6.53965661546855840980945673836, −4.88277579722730928569869211454, −4.07252440381575894570721993197, −2.76457749193270614499804207913, −1.52989913772175596187391857264, 2.53329939779784211477851566590, 3.14158265568038323933360735888, 4.70184424814225382621114513343, 5.88089921724627476531913346424, 6.69139093668403527590484348878, 8.063353523939518952228360300050, 8.445675856658394066971135195547, 9.897416078995016544542743070934, 10.15611331741557313768580052495, 11.81614004261136169350674001886

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