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Properties

Label 2-378-21.17-c3-0-26
Degree $2$
Conductor $378$
Sign $-0.996 + 0.0881i$
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 + (−9.83 − 17.0i)5-s + (9.86 − 15.6i)7-s + 7.99i·8-s + (34.0 + 19.6i)10-s + (−26.1 − 15.0i)11-s − 30.1i·13-s + (−1.42 + 37.0i)14-s + (−8 − 13.8i)16-s + (60.7 − 105. i)17-s + (70.8 − 40.8i)19-s − 78.6·20-s + 60.3·22-s + (−80.2 + 46.3i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.879 − 1.52i)5-s + (0.532 − 0.846i)7-s + 0.353i·8-s + (1.07 + 0.621i)10-s + (−0.716 − 0.413i)11-s − 0.643i·13-s + (−0.0271 + 0.706i)14-s + (−0.125 − 0.216i)16-s + (0.866 − 1.50i)17-s + (0.855 − 0.493i)19-s − 0.879·20-s + 0.584·22-s + (−0.727 + 0.419i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0881i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.996 + 0.0881i$
Analytic conductor: \(22.3027\)
Root analytic conductor: \(4.72257\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :3/2),\ -0.996 + 0.0881i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7432847399\)
\(L(\frac12)\) \(\approx\) \(0.7432847399\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.86 + 15.6i)T \)
good5 \( 1 + (9.83 + 17.0i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (26.1 + 15.0i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 30.1iT - 2.19e3T^{2} \)
17 \( 1 + (-60.7 + 105. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-70.8 + 40.8i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (80.2 - 46.3i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 42.7iT - 2.43e4T^{2} \)
31 \( 1 + (-135. - 78.3i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-131. - 228. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 277.T + 6.89e4T^{2} \)
43 \( 1 + 456.T + 7.95e4T^{2} \)
47 \( 1 + (183. + 317. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-288. - 166. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (166. - 289. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-560. + 323. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-364. + 632. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 164. iT - 3.57e5T^{2} \)
73 \( 1 + (443. + 255. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (67.5 + 117. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 714.T + 5.71e5T^{2} \)
89 \( 1 + (-651. - 1.12e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 796. iT - 9.12e5T^{2} \)
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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

−10.30744661946521265032421009992, −9.508707441605590545322401317651, −8.285961193076445905902485657725, −7.987445514742828181884272556487, −7.07701272663445172684498652177, −5.27082755231852259503455185089, −4.88711832913888243454265232983, −3.34439052972513385320983784499, −1.15785355789130399207669357461, −0.35889636175806399609151753437, 1.93087242021581357847497760357, 3.02221907305947094527095708586, 4.11111029279491679757869081931, 5.78953966378662330791603396543, 6.87536854451439865506811796973, 7.905943020526157048281526156632, 8.311394635829820767960542504495, 9.893860527280176415928223484594, 10.38019949353190893717633175889, 11.48838837436089158260926775446

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