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Properties

Label 2-378-63.40-c2-0-4
Degree $2$
Conductor $378$
Sign $0.337 - 0.941i$
Analytic cond. $10.2997$
Root an. cond. $3.20932$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + 2.00·4-s + (−1.56 + 0.905i)5-s + (6.71 − 1.98i)7-s − 2.82·8-s + (2.21 − 1.28i)10-s + (−9.25 + 16.0i)11-s + (−1.27 − 0.738i)13-s + (−9.49 + 2.80i)14-s + 4.00·16-s + (15.4 − 8.90i)17-s + (6.64 + 3.83i)19-s + (−3.13 + 1.81i)20-s + (13.0 − 22.6i)22-s + (−9.25 − 16.0i)23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.500·4-s + (−0.313 + 0.181i)5-s + (0.959 − 0.283i)7-s − 0.353·8-s + (0.221 − 0.128i)10-s + (−0.841 + 1.45i)11-s + (−0.0984 − 0.0568i)13-s + (−0.678 + 0.200i)14-s + 0.250·16-s + (0.907 − 0.523i)17-s + (0.349 + 0.202i)19-s + (−0.156 + 0.0905i)20-s + (0.595 − 1.03i)22-s + (−0.402 − 0.697i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.337 - 0.941i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ 0.337 - 0.941i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.870650 + 0.612578i\)
\(L(\frac12)\) \(\approx\) \(0.870650 + 0.612578i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 \)
7 \( 1 + (-6.71 + 1.98i)T \)
good5 \( 1 + (1.56 - 0.905i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (9.25 - 16.0i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (1.27 + 0.738i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-15.4 + 8.90i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.64 - 3.83i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (9.25 + 16.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-16.2 - 28.1i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 - 24.6iT - 961T^{2} \)
37 \( 1 + (12.8 - 22.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-48.1 - 27.8i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-13.9 - 24.1i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 42.4iT - 2.20e3T^{2} \)
53 \( 1 + (20.4 + 35.4i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 - 36.4iT - 3.48e3T^{2} \)
61 \( 1 - 110. iT - 3.72e3T^{2} \)
67 \( 1 - 81.3T + 4.48e3T^{2} \)
71 \( 1 - 115.T + 5.04e3T^{2} \)
73 \( 1 + (101. - 58.7i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 43.2T + 6.24e3T^{2} \)
83 \( 1 + (25.7 - 14.8i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (60.7 + 35.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-159. + 91.9i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−11.18534867231276374789118427840, −10.31125397163386474891510179056, −9.666783857089599172198914673504, −8.385750466413333381855989710556, −7.61002394429414139218319026878, −7.04562621268415930837921109260, −5.43703257165258102371951083940, −4.45911134613799576558517329307, −2.81192955015819543675190655248, −1.39433309017615791223976480961, 0.64688895179433454000458164639, 2.29755254554071072480159695842, 3.74980406990611519757527625049, 5.29510912049294127988184389200, 6.08493855492737421203203957977, 7.77677923444042351811991310894, 8.009036589662994460014679137918, 8.980960465468298715259071651787, 10.09855550408472730088318083544, 10.98800098799601818909872628689

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