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Properties

Label 2-378-189.101-c1-0-15
Degree $2$
Conductor $378$
Sign $0.836 + 0.548i$
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 + (0.327 + 1.70i)3-s + (−0.173 − 0.984i)4-s + (1.33 − 1.11i)5-s + (1.51 + 0.842i)6-s + (1.31 − 2.29i)7-s + (−0.866 − 0.500i)8-s + (−2.78 + 1.11i)9-s − 1.73i·10-s + (1.64 − 1.96i)11-s + (1.61 − 0.617i)12-s + (0.456 − 1.25i)13-s + (−0.910 − 2.48i)14-s + (2.33 + 1.90i)15-s + (−0.939 + 0.342i)16-s + 3.74·17-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (0.188 + 0.981i)3-s + (−0.0868 − 0.492i)4-s + (0.595 − 0.500i)5-s + (0.617 + 0.343i)6-s + (0.498 − 0.867i)7-s + (−0.306 − 0.176i)8-s + (−0.928 + 0.371i)9-s − 0.550i·10-s + (0.497 − 0.592i)11-s + (0.467 − 0.178i)12-s + (0.126 − 0.347i)13-s + (−0.243 − 0.663i)14-s + (0.603 + 0.490i)15-s + (−0.234 + 0.0855i)16-s + 0.907·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.836 + 0.548i$
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: \((2,\ 378,\ (\ :1/2),\ 0.836 + 0.548i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92997 - 0.576493i\)
\(L(\frac12)\) \(\approx\) \(1.92997 - 0.576493i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.327 - 1.70i)T \)
7 \( 1 + (-1.31 + 2.29i)T \)
good5 \( 1 + (-1.33 + 1.11i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-1.64 + 1.96i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.456 + 1.25i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 3.74T + 17T^{2} \)
19 \( 1 - 5.43iT - 19T^{2} \)
23 \( 1 + (1.55 - 4.27i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.79 + 4.92i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-3.35 + 0.590i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (1.39 - 2.42i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.82 + 2.12i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.74 - 9.92i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.51 + 8.60i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (0.290 + 0.167i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.94 - 2.16i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (10.2 + 1.81i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (7.81 - 6.55i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (6.08 - 3.51i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (12.7 - 7.37i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-9.64 - 8.09i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-10.6 + 3.87i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 17.1T + 89T^{2} \)
97 \( 1 + (-0.750 - 0.132i)T + (91.1 + 33.1i)T^{2} \)
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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

−11.30149213232296041237912426561, −10.13931702419408210919217654105, −9.903623862977555348490122323472, −8.697630985103930539680252593752, −7.76792927946681161454204765332, −5.98797927247626298784808652668, −5.28173885247056654565494017966, −4.12915919685108785727139252381, −3.32018505328522642462072023642, −1.46374249653155495855860027897, 1.91919005499487582385936602053, 3.01612003978633693072678295462, 4.77484561730021489034906347497, 5.90496003583901235716254906941, 6.63386462637124218926365037112, 7.48296789897188504652089947851, 8.568893388129290609945952602762, 9.278954391708193425287498292019, 10.67370919390212908568214450835, 11.91381013199410176884673444971

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