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Properties

Label 2-378-21.17-c3-0-23
Degree $2$
Conductor $378$
Sign $-0.362 + 0.931i$
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 + (−0.523 − 0.906i)5-s + (9.66 + 15.7i)7-s − 7.99i·8-s + (−1.81 − 1.04i)10-s + (−53.5 − 30.8i)11-s − 70.7i·13-s + (32.5 + 17.6i)14-s + (−8 − 13.8i)16-s + (7.82 − 13.5i)17-s + (−2.91 + 1.68i)19-s − 4.18·20-s − 123.·22-s + (141. − 81.7i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.0468 − 0.0811i)5-s + (0.521 + 0.853i)7-s − 0.353i·8-s + (−0.0573 − 0.0331i)10-s + (−1.46 − 0.846i)11-s − 1.50i·13-s + (0.621 + 0.337i)14-s + (−0.125 − 0.216i)16-s + (0.111 − 0.193i)17-s + (−0.0352 + 0.0203i)19-s − 0.0468·20-s − 1.19·22-s + (1.28 − 0.740i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.362 + 0.931i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.362 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.362 + 0.931i$
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: \((2,\ 378,\ (\ :3/2),\ -0.362 + 0.931i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.194463792\)
\(L(\frac12)\) \(\approx\) \(2.194463792\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.66 - 15.7i)T \)
good5 \( 1 + (0.523 + 0.906i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (53.5 + 30.8i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 70.7iT - 2.19e3T^{2} \)
17 \( 1 + (-7.82 + 13.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (2.91 - 1.68i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-141. + 81.7i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 39.3iT - 2.43e4T^{2} \)
31 \( 1 + (-200. - 115. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (115. + 200. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 404.T + 6.89e4T^{2} \)
43 \( 1 + 65.6T + 7.95e4T^{2} \)
47 \( 1 + (298. + 516. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (190. + 109. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-248. + 430. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-313. + 180. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (365. - 633. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 141. iT - 3.57e5T^{2} \)
73 \( 1 + (-688. - 397. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-530. - 919. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 101.T + 5.71e5T^{2} \)
89 \( 1 + (-333. - 577. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 588. iT - 9.12e5T^{2} \)
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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

−10.70416429998077368188971442596, −10.11924508537429694793949295533, −8.538733331890149315589187952739, −8.125324303712930516962554459926, −6.63852041166021305394814433843, −5.30551055373935553722137641536, −5.10370478280682739685581331708, −3.23360178449016922808510730940, −2.46932784176195556876353935881, −0.60096728523708094181296539614, 1.67407766796266953287403290341, 3.17231343191986796161011713375, 4.55398118887155161692143272941, 5.07097338801672927915333390011, 6.62110835354313609546064736794, 7.32815420915638739832922797213, 8.139328451168640585101030851602, 9.413543073516837861860963638318, 10.47515645939981868583843496517, 11.27333553821410480760524584059

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