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Properties

Label 2-378-21.5-c3-0-19
Degree $2$
Conductor $378$
Sign $0.926 + 0.377i$
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  + (−1.73 − i)2-s + (1.99 + 3.46i)4-s + (−4.04 + 7.00i)5-s + (17.8 − 4.81i)7-s − 7.99i·8-s + (14.0 − 8.08i)10-s + (−8.85 + 5.11i)11-s − 59.8i·13-s + (−35.7 − 9.53i)14-s + (−8 + 13.8i)16-s + (17.8 + 30.9i)17-s + (−30.4 − 17.5i)19-s − 32.3·20-s + 20.4·22-s + (−11.4 − 6.62i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.361 + 0.626i)5-s + (0.965 − 0.260i)7-s − 0.353i·8-s + (0.442 − 0.255i)10-s + (−0.242 + 0.140i)11-s − 1.27i·13-s + (−0.683 − 0.182i)14-s + (−0.125 + 0.216i)16-s + (0.255 + 0.441i)17-s + (−0.367 − 0.212i)19-s − 0.361·20-s + 0.198·22-s + (−0.103 − 0.0600i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.377i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.926 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.926 + 0.377i$
Analytic conductor: \(22.3027\)
Root analytic conductor: \(4.72257\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :3/2),\ 0.926 + 0.377i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.395919501\)
\(L(\frac12)\) \(\approx\) \(1.395919501\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 + i)T \)
3 \( 1 \)
7 \( 1 + (-17.8 + 4.81i)T \)
good5 \( 1 + (4.04 - 7.00i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (8.85 - 5.11i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 59.8iT - 2.19e3T^{2} \)
17 \( 1 + (-17.8 - 30.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (30.4 + 17.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (11.4 + 6.62i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 56.7iT - 2.43e4T^{2} \)
31 \( 1 + (-199. + 115. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-6.66 + 11.5i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 176.T + 6.89e4T^{2} \)
43 \( 1 - 354.T + 7.95e4T^{2} \)
47 \( 1 + (76.6 - 132. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-28.8 + 16.6i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (106. + 185. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-362. - 209. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-137. - 238. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 1.14e3iT - 3.57e5T^{2} \)
73 \( 1 + (-169. + 98.0i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-357. + 619. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.40e3T + 5.71e5T^{2} \)
89 \( 1 + (-296. + 514. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 325. iT - 9.12e5T^{2} \)
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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

−10.73611183639683283609072865996, −10.29282651131314790440804706581, −9.008686533442436086705550254285, −7.910378681119567989868071496358, −7.57114398211722146754470661343, −6.21477878853573299327057668642, −4.87569631159796936027830520899, −3.57370960012080113845774586400, −2.38046017837979818021304588471, −0.824972823086866897671664713245, 0.936080459239370938721896190273, 2.28072743084491838386834063131, 4.24929991286640597428153093330, 5.10524509294447383549552213215, 6.31667577451754741404675912327, 7.44611681668633423868171632738, 8.327913960183292334468394050732, 8.912038474402208277803853227504, 9.935376448459497669849737458827, 11.04364340762474531325088051129

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