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Properties

Label 2-378-189.101-c1-0-23
Degree $2$
Conductor $378$
Sign $-0.900 + 0.433i$
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 + (−0.391 − 1.68i)3-s + (−0.173 − 0.984i)4-s + (1.01 − 0.849i)5-s + (−1.54 − 0.784i)6-s + (−1.01 − 2.44i)7-s + (−0.866 − 0.500i)8-s + (−2.69 + 1.32i)9-s − 1.32i·10-s + (−2.07 + 2.47i)11-s + (−1.59 + 0.678i)12-s + (0.846 − 2.32i)13-s + (−2.52 − 0.790i)14-s + (−1.82 − 1.37i)15-s + (−0.939 + 0.342i)16-s + 5.44·17-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (−0.225 − 0.974i)3-s + (−0.0868 − 0.492i)4-s + (0.452 − 0.379i)5-s + (−0.630 − 0.320i)6-s + (−0.384 − 0.923i)7-s + (−0.306 − 0.176i)8-s + (−0.897 + 0.440i)9-s − 0.417i·10-s + (−0.626 + 0.747i)11-s + (−0.460 + 0.195i)12-s + (0.234 − 0.644i)13-s + (−0.674 − 0.211i)14-s + (−0.472 − 0.355i)15-s + (−0.234 + 0.0855i)16-s + 1.31·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.433i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.900 + 0.433i$
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: \((2,\ 378,\ (\ :1/2),\ -0.900 + 0.433i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.324089 - 1.41987i\)
\(L(\frac12)\) \(\approx\) \(0.324089 - 1.41987i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.391 + 1.68i)T \)
7 \( 1 + (1.01 + 2.44i)T \)
good5 \( 1 + (-1.01 + 0.849i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (2.07 - 2.47i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.846 + 2.32i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 5.44T + 17T^{2} \)
19 \( 1 - 3.50iT - 19T^{2} \)
23 \( 1 + (-2.72 + 7.48i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.74 + 4.80i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (9.30 - 1.64i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-4.35 + 7.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.84 - 2.49i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.724 + 4.11i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.0358 + 0.203i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-4.50 - 2.60i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.79 - 2.83i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-6.71 - 1.18i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (4.76 - 3.99i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-13.2 + 7.66i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.87 + 3.97i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.07 + 1.73i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (1.52 - 0.554i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + (4.79 + 0.845i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.92715516994810730793402288917, −10.32989502454217978734727572437, −9.328236183542865684926081566457, −7.931494039155077718299651570247, −7.21529328164028248955435006657, −5.95973076094349925829818384319, −5.22635459843891252706381702377, −3.74216155240082380963704907455, −2.32673543610218934736710927826, −0.898632703338837109872492577684, 2.78116261292293210222430296141, 3.70147337457321497104976213514, 5.30966722414030005724124336443, 5.64890413947216575325933620480, 6.74533490962564754730224519873, 8.103403836401436809091050067442, 9.163809392213534075034461505449, 9.726920120952644302866905659689, 10.97872125287084897242603862017, 11.59790328758213796571798761159

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