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Properties

Label 2-378-7.2-c3-0-21
Degree $2$
Conductor $378$
Sign $0.743 + 0.668i$
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 + (8.97 − 15.5i)5-s + (13.2 − 12.9i)7-s + 7.99·8-s + (17.9 + 31.0i)10-s + (19.8 + 34.3i)11-s + 28.4·13-s + (9.22 + 35.8i)14-s + (−8 + 13.8i)16-s + (−7.38 − 12.7i)17-s + (−33.9 + 58.7i)19-s − 71.8·20-s − 79.3·22-s + (88.8 − 153. i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.802 − 1.39i)5-s + (0.714 − 0.699i)7-s + 0.353·8-s + (0.567 + 0.983i)10-s + (0.543 + 0.941i)11-s + 0.606·13-s + (0.176 + 0.684i)14-s + (−0.125 + 0.216i)16-s + (−0.105 − 0.182i)17-s + (−0.409 + 0.709i)19-s − 0.802·20-s − 0.769·22-s + (0.805 − 1.39i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.743 + 0.668i$
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: \((2,\ 378,\ (\ :3/2),\ 0.743 + 0.668i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.989858941\)
\(L(\frac12)\) \(\approx\) \(1.989858941\)
\(L(\frac{5}{2})\) 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 \( 1 + (-13.2 + 12.9i)T \)
good5 \( 1 + (-8.97 + 15.5i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-19.8 - 34.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 28.4T + 2.19e3T^{2} \)
17 \( 1 + (7.38 + 12.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (33.9 - 58.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-88.8 + 153. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 209.T + 2.43e4T^{2} \)
31 \( 1 + (143. + 249. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (183. - 318. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 294.T + 6.89e4T^{2} \)
43 \( 1 + 348.T + 7.95e4T^{2} \)
47 \( 1 + (-119. + 206. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-183. - 318. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (127. + 221. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-149. + 258. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-499. - 864. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 659.T + 3.57e5T^{2} \)
73 \( 1 + (319. + 553. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-338. + 586. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 830.T + 5.71e5T^{2} \)
89 \( 1 + (-231. + 401. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 373.T + 9.12e5T^{2} \)
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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

−10.51862471364645784154677236258, −9.798032112994361371084318716609, −8.796830121691145572581016244090, −8.272461430407586486873919562279, −7.05356060542782117787850829788, −6.04977450678913865625419373641, −4.88823318786455086404210222595, −4.29614490286990680367411441461, −1.82673457909873439322690772013, −0.863290911275095227424949431708, 1.44716004557945228148760743208, 2.63444660352440886679625782307, 3.56917847760746980526872279761, 5.29161088483025461875560103036, 6.29739892921468591701966505083, 7.25329041063390194424209690372, 8.610296578027111274212019218605, 9.164648955201386346950258960267, 10.37725056137066969787754149735, 11.05370990711578911637909820939

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