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Properties

Label 2-378-7.2-c3-0-15
Degree $2$
Conductor $378$
Sign $0.999 + 0.0290i$
Analytic cond. $22.3027$
Root an. cond. $4.72257$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (−8.00 + 13.8i)5-s + (1.70 − 18.4i)7-s + 7.99·8-s + (−16.0 − 27.7i)10-s + (7.89 + 13.6i)11-s − 50.4·13-s + (30.2 + 21.4i)14-s + (−8 + 13.8i)16-s + (−18.4 − 32.0i)17-s + (44.6 − 77.4i)19-s + 64.0·20-s − 31.5·22-s + (30.6 − 53.1i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.716 + 1.24i)5-s + (0.0922 − 0.995i)7-s + 0.353·8-s + (−0.506 − 0.877i)10-s + (0.216 + 0.374i)11-s − 1.07·13-s + (0.577 + 0.408i)14-s + (−0.125 + 0.216i)16-s + (−0.263 − 0.456i)17-s + (0.539 − 0.934i)19-s + 0.716·20-s − 0.306·22-s + (0.278 − 0.481i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0290i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.999 + 0.0290i$
Analytic conductor: \(22.3027\)
Root analytic conductor: \(4.72257\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :3/2),\ 0.999 + 0.0290i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.011023380\)
\(L(\frac12)\) \(\approx\) \(1.011023380\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
3 \( 1 \)
7 \( 1 + (-1.70 + 18.4i)T \)
good5 \( 1 + (8.00 - 13.8i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-7.89 - 13.6i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 50.4T + 2.19e3T^{2} \)
17 \( 1 + (18.4 + 32.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-44.6 + 77.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-30.6 + 53.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 126.T + 2.43e4T^{2} \)
31 \( 1 + (-129. - 225. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-123. + 213. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 516.T + 6.89e4T^{2} \)
43 \( 1 + 34.2T + 7.95e4T^{2} \)
47 \( 1 + (-182. + 316. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (23.5 + 40.8i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-358. - 620. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-348. + 603. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (55.3 + 95.8i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 607.T + 3.57e5T^{2} \)
73 \( 1 + (-140. - 242. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-441. + 764. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 63.1T + 5.71e5T^{2} \)
89 \( 1 + (337. - 585. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 976.T + 9.12e5T^{2} \)
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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

−10.84963297767316778104969555654, −10.05659719168898427638075979609, −9.117008112509984092065336005115, −7.70823782598669347944607739786, −7.21170108384357576032515045761, −6.64500337473951383737464926251, −5.02436463979391563711410775962, −3.97335346672927476424184483264, −2.61968211464236578098293686241, −0.51422186877649847411972184561, 0.957360864860469165726369789507, 2.40246472371457043392049433565, 3.86999114301430309552949000768, 4.89035770542477010951312591479, 5.92004538944911059551076229761, 7.66718642315709133639249986275, 8.251011846351927019330803111096, 9.203968329697177204014082720413, 9.761416498923706114011744213613, 11.23605477161504199018082742376

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