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Properties

Label 2-378-21.17-c3-0-20
Degree $2$
Conductor $378$
Sign $0.196 + 0.980i$
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 + (4.16 + 7.20i)5-s + (−16.6 − 8.10i)7-s − 7.99i·8-s + (14.4 + 8.32i)10-s + (20.0 + 11.5i)11-s − 39.4i·13-s + (−36.9 + 2.61i)14-s + (−8 − 13.8i)16-s + (50.8 − 88.0i)17-s + (28.7 − 16.5i)19-s + 33.2·20-s + 46.2·22-s + (25.0 − 14.4i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.372 + 0.644i)5-s + (−0.899 − 0.437i)7-s − 0.353i·8-s + (0.455 + 0.263i)10-s + (0.549 + 0.317i)11-s − 0.841i·13-s + (−0.705 + 0.0499i)14-s + (−0.125 − 0.216i)16-s + (0.725 − 1.25i)17-s + (0.346 − 0.200i)19-s + 0.372·20-s + 0.448·22-s + (0.226 − 0.131i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.196 + 0.980i$
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: \((2,\ 378,\ (\ :3/2),\ 0.196 + 0.980i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.572905028\)
\(L(\frac12)\) \(\approx\) \(2.572905028\)
\(L(\frac{5}{2})\) 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 \( 1 + (16.6 + 8.10i)T \)
good5 \( 1 + (-4.16 - 7.20i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-20.0 - 11.5i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 39.4iT - 2.19e3T^{2} \)
17 \( 1 + (-50.8 + 88.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-28.7 + 16.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-25.0 + 14.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 177. iT - 2.43e4T^{2} \)
31 \( 1 + (-135. - 78.1i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (9.16 + 15.8i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 123.T + 6.89e4T^{2} \)
43 \( 1 - 242.T + 7.95e4T^{2} \)
47 \( 1 + (31.7 + 55.0i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-54.9 - 31.6i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-215. + 373. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (749. - 432. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-93.3 + 161. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 1.16e3iT - 3.57e5T^{2} \)
73 \( 1 + (667. + 385. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-323. - 560. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 231.T + 5.71e5T^{2} \)
89 \( 1 + (129. + 224. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.35e3iT - 9.12e5T^{2} \)
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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

−10.64355673761719140913961205243, −10.01760296838228844147801018223, −9.270871131590227829974441973711, −7.65204608683697309761514753093, −6.75600292808842497561188578915, −5.94425681312387192258853269836, −4.72437088515226861827764373681, −3.40342050597453539299223059364, −2.62049293851507172351493964585, −0.76536406029214100608963278230, 1.46933756718226387146789854759, 3.11548449961764763740350149673, 4.18049572860788732293531344835, 5.46887491105687409914567503515, 6.17476845236414732607063335542, 7.12970622082253663371852828954, 8.478377812549456332880307844949, 9.170053496683885464868453604835, 10.11389809620389871121755856583, 11.37812112612399974028744069026

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