Properties

Label 2-378-189.83-c1-0-13
Degree $2$
Conductor $378$
Sign $0.979 + 0.199i$
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 + (−1.64 + 0.540i)3-s + (0.939 + 0.342i)4-s + (0.881 − 0.739i)5-s + (−1.71 + 0.246i)6-s + (−0.336 − 2.62i)7-s + (0.866 + 0.5i)8-s + (2.41 − 1.77i)9-s + (0.996 − 0.575i)10-s + (0.595 − 0.709i)11-s + (−1.73 − 0.0550i)12-s + (4.25 − 0.749i)13-s + (0.124 − 2.64i)14-s + (−1.05 + 1.69i)15-s + (0.766 + 0.642i)16-s + (2.28 + 3.95i)17-s + ⋯
L(s)  = 1  + (0.696 + 0.122i)2-s + (−0.950 + 0.311i)3-s + (0.469 + 0.171i)4-s + (0.394 − 0.330i)5-s + (−0.699 + 0.100i)6-s + (−0.127 − 0.991i)7-s + (0.306 + 0.176i)8-s + (0.805 − 0.592i)9-s + (0.315 − 0.181i)10-s + (0.179 − 0.214i)11-s + (−0.499 − 0.0158i)12-s + (1.17 − 0.207i)13-s + (0.0332 − 0.706i)14-s + (−0.271 + 0.437i)15-s + (0.191 + 0.160i)16-s + (0.553 + 0.958i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.979 + 0.199i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.979 + 0.199i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68693 - 0.169716i\)
\(L(\frac12)\) \(\approx\) \(1.68693 - 0.169716i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 - 0.540i)T \)
7 \( 1 + (0.336 + 2.62i)T \)
good5 \( 1 + (-0.881 + 0.739i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-0.595 + 0.709i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-4.25 + 0.749i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-2.28 - 3.95i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.687 + 0.397i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.46 + 4.01i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.10 - 0.195i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (-1.11 + 3.07i)T + (-23.7 - 19.9i)T^{2} \)
37 \( 1 + (5.27 + 9.13i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.15 - 6.56i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (2.26 + 1.89i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-1.43 + 0.522i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 - 6.12iT - 53T^{2} \)
59 \( 1 + (-0.539 + 0.453i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (2.12 + 5.83i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.0261 - 0.148i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (13.9 - 8.05i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.65 + 3.26i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.17 - 12.3i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (2.26 - 12.8i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (6.04 - 10.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.91 + 4.66i)T + (-16.8 - 95.5i)T^{2} \)
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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

−11.16743936406412043718252732069, −10.72050723090199993885663500553, −9.799034357817417917178343508342, −8.523277276878144433555056294866, −7.23980202863669173929254372575, −6.26189627198707309610604929036, −5.59018024630883108434756811738, −4.37660203840667252278236896740, −3.57177506903511634709695011677, −1.25300665112811944624295830200, 1.66620009950497562630093771019, 3.14476002306736864745971061018, 4.69981204465554005282071420989, 5.67978328996811767079859564251, 6.31948968094841739169646279727, 7.24157433510216101067823117377, 8.635374654040669663412596264744, 9.856459179671618537910574877442, 10.67822008948551551598700311966, 11.81309550467848404638861926350

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