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Properties

Label 2-378-9.2-c2-0-7
Degree $2$
Conductor $378$
Sign $0.990 + 0.140i$
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  + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s + (3.60 − 2.08i)5-s + (−1.32 + 2.29i)7-s − 2.82i·8-s − 5.89·10-s + (5.60 + 3.23i)11-s + (7.36 + 12.7i)13-s + (3.24 − 1.87i)14-s + (−2.00 + 3.46i)16-s − 18.6i·17-s + 15.6·19-s + (7.21 + 4.16i)20-s + (−4.57 − 7.92i)22-s + (−17.1 + 9.88i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.721 − 0.416i)5-s + (−0.188 + 0.327i)7-s − 0.353i·8-s − 0.589·10-s + (0.509 + 0.294i)11-s + (0.566 + 0.980i)13-s + (0.231 − 0.133i)14-s + (−0.125 + 0.216i)16-s − 1.09i·17-s + 0.826·19-s + (0.360 + 0.208i)20-s + (−0.208 − 0.360i)22-s + (−0.744 + 0.429i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.990 + 0.140i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ 0.990 + 0.140i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.46929 - 0.103616i\)
\(L(\frac12)\) \(\approx\) \(1.46929 - 0.103616i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 \)
7 \( 1 + (1.32 - 2.29i)T \)
good5 \( 1 + (-3.60 + 2.08i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-5.60 - 3.23i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-7.36 - 12.7i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 18.6iT - 289T^{2} \)
19 \( 1 - 15.6T + 361T^{2} \)
23 \( 1 + (17.1 - 9.88i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-9.30 - 5.37i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-11.1 - 19.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 70.8T + 1.36e3T^{2} \)
41 \( 1 + (-7.29 + 4.20i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-24.9 + 43.2i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-62.5 - 36.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 80.3iT - 2.80e3T^{2} \)
59 \( 1 + (-22.8 + 13.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-32.7 + 56.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (11.9 + 20.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 111. iT - 5.04e3T^{2} \)
73 \( 1 - 13.3T + 5.32e3T^{2} \)
79 \( 1 + (59.8 - 103. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (10.0 + 5.81i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 83.5iT - 7.92e3T^{2} \)
97 \( 1 + (84.0 - 145. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.24887684313275397087984211907, −9.843846703783059457066824740056, −9.448580836547781876728424964226, −8.669652816273646442178986961784, −7.43152384896121511162751103057, −6.43594044724590171067686989463, −5.33318801689837646667033099345, −3.96222936158074687021620418090, −2.46133187763108865898193508932, −1.20071831140843240879061462609, 1.02456638777289451282135275813, 2.64835796675551464103701984068, 4.11852856651082033767553042107, 5.92530823384303747203768460208, 6.16023785913294268093569380115, 7.53120623151958335830367131099, 8.331483434988123886471541209718, 9.422346604973782166507699136590, 10.21361517191184966699476142544, 10.81826657377384879437472122155

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