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Properties

Label 2-378-9.2-c2-0-8
Degree $2$
Conductor $378$
Sign $0.648 - 0.760i$
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 + (3.47 − 2.00i)5-s + (−1.32 + 2.29i)7-s + 2.82i·8-s + 5.67·10-s + (2.91 + 1.68i)11-s + (10.3 + 18.0i)13-s + (−3.24 + 1.87i)14-s + (−2.00 + 3.46i)16-s − 9.33i·17-s + 9.84·19-s + (6.94 + 4.01i)20-s + (2.37 + 4.12i)22-s + (31.0 − 17.9i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.694 − 0.401i)5-s + (−0.188 + 0.327i)7-s + 0.353i·8-s + 0.567·10-s + (0.264 + 0.152i)11-s + (0.799 + 1.38i)13-s + (−0.231 + 0.133i)14-s + (−0.125 + 0.216i)16-s − 0.548i·17-s + 0.518·19-s + (0.347 + 0.200i)20-s + (0.108 + 0.187i)22-s + (1.34 − 0.779i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 - 0.760i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.648 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.648 - 0.760i$
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: \((2,\ 378,\ (\ :1),\ 0.648 - 0.760i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.49966 + 1.15344i\)
\(L(\frac12)\) \(\approx\) \(2.49966 + 1.15344i\)
\(L(2)\) 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 \( 1 + (-3.47 + 2.00i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-2.91 - 1.68i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-10.3 - 18.0i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 9.33iT - 289T^{2} \)
19 \( 1 - 9.84T + 361T^{2} \)
23 \( 1 + (-31.0 + 17.9i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-24.8 - 14.3i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-1.68 - 2.92i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 65.0T + 1.36e3T^{2} \)
41 \( 1 + (21.0 - 12.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-12.3 + 21.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-40.6 - 23.4i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 63.1iT - 2.80e3T^{2} \)
59 \( 1 + (43.7 - 25.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (18.8 - 32.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-26.8 - 46.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 84.1iT - 5.04e3T^{2} \)
73 \( 1 + 119.T + 5.32e3T^{2} \)
79 \( 1 + (-73.0 + 126. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (84.8 + 48.9i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 78.6iT - 7.92e3T^{2} \)
97 \( 1 + (-45.7 + 79.2i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.52201305438774633712730484818, −10.39557395693587586544684800500, −9.116015522871299981645636042899, −8.781239943225638360943660084380, −7.17629307681480542942707855328, −6.46506016536363436748048486590, −5.40850238285064368541898358097, −4.50236727793514555952765100588, −3.13011183150250561542899573282, −1.62307147224541956192661128266, 1.19478080712638267689701701544, 2.82442534805527256169244537150, 3.74747794125553129263510950092, 5.24318983901964675278150572866, 6.04005037019074324667682516535, 6.99627314089205210299415605717, 8.237979514216420356031574237408, 9.422226383865387005424880126152, 10.41581790807204895784738881275, 10.83515574527452088248328706131

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