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Properties

Label 2-378-189.101-c1-0-6
Degree $2$
Conductor $378$
Sign $-0.610 - 0.792i$
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 + (−1.68 + 0.390i)3-s + (−0.173 − 0.984i)4-s + (−0.278 + 0.234i)5-s + (0.785 − 1.54i)6-s + (1.49 + 2.17i)7-s + (0.866 + 0.500i)8-s + (2.69 − 1.31i)9-s − 0.364i·10-s + (2.84 − 3.38i)11-s + (0.677 + 1.59i)12-s + (−1.20 + 3.31i)13-s + (−2.63 − 0.252i)14-s + (0.379 − 0.504i)15-s + (−0.939 + 0.342i)16-s − 5.47·17-s + ⋯
L(s)  = 1  + (−0.454 + 0.541i)2-s + (−0.974 + 0.225i)3-s + (−0.0868 − 0.492i)4-s + (−0.124 + 0.104i)5-s + (0.320 − 0.630i)6-s + (0.566 + 0.823i)7-s + (0.306 + 0.176i)8-s + (0.898 − 0.439i)9-s − 0.115i·10-s + (0.857 − 1.02i)11-s + (0.195 + 0.460i)12-s + (−0.334 + 0.919i)13-s + (−0.703 − 0.0674i)14-s + (0.0979 − 0.130i)15-s + (−0.234 + 0.0855i)16-s − 1.32·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 - 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.610 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.610 - 0.792i$
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: \((2,\ 378,\ (\ :1/2),\ -0.610 - 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.294928 + 0.599238i\)
\(L(\frac12)\) \(\approx\) \(0.294928 + 0.599238i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 - 0.390i)T \)
7 \( 1 + (-1.49 - 2.17i)T \)
good5 \( 1 + (0.278 - 0.234i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-2.84 + 3.38i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (1.20 - 3.31i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 5.47T + 17T^{2} \)
19 \( 1 - 5.51iT - 19T^{2} \)
23 \( 1 + (1.70 - 4.68i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.217 + 0.596i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-4.90 + 0.865i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (4.30 - 7.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.52 - 1.28i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.51 + 8.58i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.967 - 5.48i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (7.94 + 4.58i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.48 - 3.45i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (5.46 + 0.962i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-6.89 + 5.78i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (3.24 - 1.87i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.79 + 2.19i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-12.9 - 10.8i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-0.538 + 0.196i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 0.706T + 89T^{2} \)
97 \( 1 + (-12.0 - 2.13i)T + (91.1 + 33.1i)T^{2} \)
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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

−11.54397587641031733402271187274, −10.95822103916641123740241744704, −9.698997950190981346400380692133, −8.999703485003612694069715490823, −7.994471432357304607065299760489, −6.73338898832530777763646384502, −6.05576347237847537627290519753, −5.09857480327732194787839533430, −3.91775718669682071274890810470, −1.63992322921739993945467432882, 0.61478185683101944831522809120, 2.17792168321731687860447002870, 4.25587479467670968295105210063, 4.79750288805043858600571121186, 6.51225678116884957796855689444, 7.20716247741222352911771255505, 8.218242031291461913254771762315, 9.427450891141532706155837838589, 10.41154885369867251474172667095, 10.95637539343925736726786422551

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