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Properties

Label 2-378-9.7-c3-0-14
Degree $2$
Conductor $378$
Sign $-0.914 + 0.405i$
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 + (0.721 + 1.24i)5-s + (3.5 − 6.06i)7-s − 7.99·8-s + 2.88·10-s + (8.37 − 14.5i)11-s + (−14.0 − 24.2i)13-s + (−7 − 12.1i)14-s + (−8 + 13.8i)16-s + 95.7·17-s − 145.·19-s + (2.88 − 4.99i)20-s + (−16.7 − 29.0i)22-s + (−32.3 − 56.0i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.0645 + 0.111i)5-s + (0.188 − 0.327i)7-s − 0.353·8-s + 0.0912·10-s + (0.229 − 0.397i)11-s + (−0.298 − 0.517i)13-s + (−0.133 − 0.231i)14-s + (−0.125 + 0.216i)16-s + 1.36·17-s − 1.75·19-s + (0.0322 − 0.0558i)20-s + (−0.162 − 0.281i)22-s + (−0.293 − 0.508i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.405i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.914 + 0.405i$
Analytic conductor: \(22.3027\)
Root analytic conductor: \(4.72257\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :3/2),\ -0.914 + 0.405i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.546492057\)
\(L(\frac12)\) \(\approx\) \(1.546492057\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.5 + 6.06i)T \)
good5 \( 1 + (-0.721 - 1.24i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-8.37 + 14.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (14.0 + 24.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 95.7T + 4.91e3T^{2} \)
19 \( 1 + 145.T + 6.85e3T^{2} \)
23 \( 1 + (32.3 + 56.0i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (34.0 - 58.9i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (134. + 233. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 117.T + 5.06e4T^{2} \)
41 \( 1 + (114. + 198. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (51.7 - 89.7i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-52.0 + 90.1i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 602.T + 1.48e5T^{2} \)
59 \( 1 + (-26.9 - 46.6i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-251. + 434. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-88.5 - 153. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 398.T + 3.57e5T^{2} \)
73 \( 1 + 520.T + 3.89e5T^{2} \)
79 \( 1 + (-276. + 478. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (678. - 1.17e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 124.T + 7.04e5T^{2} \)
97 \( 1 + (-875. + 1.51e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.57015934121513846043106106467, −9.950344003564166167236619504206, −8.740017440653369288983794804584, −7.84034690812608971519929760508, −6.54584700259397647051032713609, −5.56085061991528538993006041103, −4.39157467760278231066282747268, −3.34411234360318503053717577858, −2.02147666517202729179636933391, −0.45689479608477871996306945224, 1.75524533632562382693006999037, 3.37702222496542764486036966361, 4.58618225995195422090324001528, 5.52073400366595016510506925370, 6.57035778532183042762825239301, 7.49002341012487987278988446684, 8.493030391927614321595860124474, 9.328833171326256693615560448196, 10.36045164148328982673548698121, 11.51022539764571224089745654270

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