Properties

Label 2-378-21.17-c3-0-24
Degree $2$
Conductor $378$
Sign $-0.327 + 0.944i$
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 + (−3.89 − 6.75i)5-s + (15.4 − 10.2i)7-s − 7.99i·8-s + (−13.5 − 7.79i)10-s + (44.0 + 25.4i)11-s − 50.9i·13-s + (16.4 − 33.1i)14-s + (−8 − 13.8i)16-s + (−4.37 + 7.57i)17-s + (−87.9 + 50.7i)19-s − 31.1·20-s + 101.·22-s + (36.9 − 21.3i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.348 − 0.603i)5-s + (0.832 − 0.553i)7-s − 0.353i·8-s + (−0.427 − 0.246i)10-s + (1.20 + 0.696i)11-s − 1.08i·13-s + (0.314 − 0.633i)14-s + (−0.125 − 0.216i)16-s + (−0.0623 + 0.108i)17-s + (−1.06 + 0.613i)19-s − 0.348·20-s + 0.985·22-s + (0.335 − 0.193i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.327 + 0.944i$
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: \((2,\ 378,\ (\ :3/2),\ -0.327 + 0.944i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.699276794\)
\(L(\frac12)\) \(\approx\) \(2.699276794\)
\(L(\frac{5}{2})\) 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 \( 1 + (-15.4 + 10.2i)T \)
good5 \( 1 + (3.89 + 6.75i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-44.0 - 25.4i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 50.9iT - 2.19e3T^{2} \)
17 \( 1 + (4.37 - 7.57i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (87.9 - 50.7i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-36.9 + 21.3i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 218. iT - 2.43e4T^{2} \)
31 \( 1 + (87.6 + 50.5i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (137. + 237. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 6.40T + 6.89e4T^{2} \)
43 \( 1 + 262.T + 7.95e4T^{2} \)
47 \( 1 + (-53.2 - 92.2i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-555. - 320. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-52.5 + 91.0i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-471. + 272. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-109. + 189. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 542. iT - 3.57e5T^{2} \)
73 \( 1 + (778. + 449. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-561. - 971. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 855.T + 5.71e5T^{2} \)
89 \( 1 + (-658. - 1.14e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.24e3iT - 9.12e5T^{2} \)
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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

−10.77890427507049964438992274963, −9.982277398718955121508287582710, −8.751995697780279031043702759196, −7.88265606395399675313970082775, −6.78758720570955412048070970145, −5.58040528886658091781258839078, −4.45000349425748072376271078677, −3.86856200418758039278688303796, −2.06351118202348230513184749915, −0.78548854683632023269816763050, 1.72057823546993107522913540217, 3.21117268750603995624870074028, 4.28493262771683814091257761980, 5.33981702579027444188087358189, 6.60494787088123424126468982246, 7.09702149844723909188316247687, 8.573621856497993330733696483317, 8.976705377545420182981843686451, 10.63797651290749873051326461353, 11.52437856151469368275849640036

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