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Properties

Label 2-378-63.34-c2-0-9
Degree $2$
Conductor $378$
Sign $0.0677 + 0.997i$
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  + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s + (−8.58 + 4.95i)5-s + (6.88 + 1.25i)7-s − 2.82·8-s + 14.0i·10-s + (6.74 − 11.6i)11-s + (10.5 − 6.07i)13-s + (6.40 − 7.54i)14-s + (−2.00 + 3.46i)16-s − 20.1i·17-s − 4.66i·19-s + (17.1 + 9.91i)20-s + (−9.53 − 16.5i)22-s + (0.0880 + 0.152i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−1.71 + 0.991i)5-s + (0.983 + 0.179i)7-s − 0.353·8-s + 1.40i·10-s + (0.612 − 1.06i)11-s + (0.809 − 0.467i)13-s + (0.457 − 0.539i)14-s + (−0.125 + 0.216i)16-s − 1.18i·17-s − 0.245i·19-s + (0.858 + 0.495i)20-s + (−0.433 − 0.750i)22-s + (0.00382 + 0.00662i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0677 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0677 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.0677 + 0.997i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ 0.0677 + 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.11250 - 1.03947i\)
\(L(\frac12)\) \(\approx\) \(1.11250 - 1.03947i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 \)
7 \( 1 + (-6.88 - 1.25i)T \)
good5 \( 1 + (8.58 - 4.95i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-6.74 + 11.6i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-10.5 + 6.07i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 20.1iT - 289T^{2} \)
19 \( 1 + 4.66iT - 361T^{2} \)
23 \( 1 + (-0.0880 - 0.152i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-7.01 + 12.1i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-0.0205 + 0.0118i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 2.42T + 1.36e3T^{2} \)
41 \( 1 + (-9.48 + 5.47i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (3.53 - 6.11i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-39.7 - 22.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 57.1T + 2.80e3T^{2} \)
59 \( 1 + (41.9 - 24.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (34.2 + 19.7i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (45.1 + 78.2i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 80.6T + 5.04e3T^{2} \)
73 \( 1 + 17.8iT - 5.32e3T^{2} \)
79 \( 1 + (-10.1 + 17.6i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (25.2 + 14.5i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 81.5iT - 7.92e3T^{2} \)
97 \( 1 + (94.2 + 54.4i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.18763844523711434706312639127, −10.54910892163421172427454461573, −8.951873974848498952492021703548, −8.142584981119853176114483225467, −7.26905274941175735595250920554, −6.06020228963235647581062967098, −4.65589206895110914049316165249, −3.69747784526485098288295895620, −2.82047872501035751848610109850, −0.71860743580311654316701612597, 1.34960822580336895444150865082, 3.92603578675148048134621451760, 4.24633109552575599938757101282, 5.30748355242698932028296972549, 6.81943916198312497507086359642, 7.71057035159873880483536110379, 8.393531303867751041639931797379, 9.057740937673175702130784282343, 10.70145585510413799147944155783, 11.66038607739991479125598878685

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