Properties

Label 2-378-27.4-c1-0-4
Degree $2$
Conductor $378$
Sign $0.361 - 0.932i$
Analytic cond. $3.01834$
Root an. cond. $1.73733$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 − 0.342i)2-s + (1.22 + 1.22i)3-s + (0.766 + 0.642i)4-s + (−0.0262 + 0.148i)5-s + (−0.736 − 1.56i)6-s + (−0.766 + 0.642i)7-s + (−0.500 − 0.866i)8-s + (0.0176 + 2.99i)9-s + (0.0755 − 0.130i)10-s + (0.400 + 2.27i)11-s + (0.156 + 1.72i)12-s + (3.71 − 1.35i)13-s + (0.939 − 0.342i)14-s + (−0.213 + 0.150i)15-s + (0.173 + 0.984i)16-s + (−2.15 + 3.73i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (0.709 + 0.705i)3-s + (0.383 + 0.321i)4-s + (−0.0117 + 0.0665i)5-s + (−0.300 − 0.639i)6-s + (−0.289 + 0.242i)7-s + (−0.176 − 0.306i)8-s + (0.00586 + 0.999i)9-s + (0.0238 − 0.0413i)10-s + (0.120 + 0.684i)11-s + (0.0450 + 0.497i)12-s + (1.03 − 0.375i)13-s + (0.251 − 0.0914i)14-s + (−0.0552 + 0.0389i)15-s + (0.0434 + 0.246i)16-s + (−0.522 + 0.905i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.361 - 0.932i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.361 - 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.991616 + 0.679452i\)
\(L(\frac12)\) \(\approx\) \(0.991616 + 0.679452i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 0.342i)T \)
3 \( 1 + (-1.22 - 1.22i)T \)
7 \( 1 + (0.766 - 0.642i)T \)
good5 \( 1 + (0.0262 - 0.148i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-0.400 - 2.27i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-3.71 + 1.35i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (2.15 - 3.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.94 - 3.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.37 + 4.50i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (0.393 + 0.143i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-0.824 - 0.692i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (1.42 - 2.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.55 - 0.565i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.156 - 0.885i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-3.47 + 2.91i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + (-1.29 + 7.34i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-8.81 + 7.39i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-5.44 + 1.98i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (2.74 - 4.75i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.41 + 5.91i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (13.0 + 4.75i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (12.5 + 4.57i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-1.95 - 3.38i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.21 + 12.5i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.23524181679702600247445713597, −10.33315307773947694497896327072, −9.850742108500759667695176356682, −8.654010602050425558198708671086, −8.304417451989979406093840608306, −7.01920925040704332571029851082, −5.80463018334417399145080174730, −4.27716647623819413668117135126, −3.28355238290177368491226948580, −1.93136760735043808730805178594, 0.989549480616510056379259740138, 2.60227951228864047575455702461, 3.88681675469611882046288446366, 5.71798279460458549893239162915, 6.73895725151944372552906515913, 7.42604154307823395169532700596, 8.591882076232718785211969321472, 9.014416424904675843953910354083, 10.05055781794051364438020736101, 11.25319982224869422200491196196

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