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Properties

Label 2-378-21.5-c3-0-30
Degree $2$
Conductor $378$
Sign $-0.874 - 0.484i$
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 + (2.91 − 5.05i)5-s + (−2.66 − 18.3i)7-s − 7.99i·8-s + (−10.1 + 5.83i)10-s + (−29.3 + 16.9i)11-s − 55.8i·13-s + (−13.7 + 34.4i)14-s + (−8 + 13.8i)16-s + (35.2 + 60.9i)17-s + (−27.4 − 15.8i)19-s + 23.3·20-s + 67.7·22-s + (−147. − 84.9i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.260 − 0.451i)5-s + (−0.144 − 0.989i)7-s − 0.353i·8-s + (−0.319 + 0.184i)10-s + (−0.803 + 0.464i)11-s − 1.19i·13-s + (−0.261 + 0.656i)14-s + (−0.125 + 0.216i)16-s + (0.502 + 0.870i)17-s + (−0.331 − 0.191i)19-s + 0.260·20-s + 0.656·22-s + (−1.33 − 0.770i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.874 - 0.484i$
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: \((2,\ 378,\ (\ :3/2),\ -0.874 - 0.484i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3037293003\)
\(L(\frac12)\) \(\approx\) \(0.3037293003\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.66 + 18.3i)T \)
good5 \( 1 + (-2.91 + 5.05i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (29.3 - 16.9i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 55.8iT - 2.19e3T^{2} \)
17 \( 1 + (-35.2 - 60.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (27.4 + 15.8i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (147. + 84.9i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 141. iT - 2.43e4T^{2} \)
31 \( 1 + (-173. + 100. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-73.3 + 127. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 325.T + 6.89e4T^{2} \)
43 \( 1 + 334.T + 7.95e4T^{2} \)
47 \( 1 + (-14.3 + 24.7i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (483. - 279. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-55.7 - 96.5i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-134. - 77.3i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (297. + 515. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 542. iT - 3.57e5T^{2} \)
73 \( 1 + (469. - 270. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (602. - 1.04e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 82.6T + 5.71e5T^{2} \)
89 \( 1 + (758. - 1.31e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 960. iT - 9.12e5T^{2} \)
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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

−10.27747387208009170831183311898, −9.815570392645472699078652081189, −8.400414403006403510781318897550, −7.88115334835195406710799561969, −6.76892821914153910008371699993, −5.50215760573314171838882865765, −4.25979039950962538267765924948, −2.98293894457159807163138680667, −1.47009856569884913341519550854, −0.12407345246198002838377148758, 1.91227970237136470154901766268, 3.03201386326309063762743375210, 4.83890693444946822708871502186, 5.96495138387371508204163963105, 6.65471424460162225800487556970, 7.902606691788407018151941141624, 8.640264069170264911843335105748, 9.702778463217390855613345987411, 10.24161916363805316710851910076, 11.57084651827039672906506682976

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