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Properties

Label 2-378-21.11-c2-0-18
Degree $2$
Conductor $378$
Sign $-0.0692 + 0.997i$
Analytic cond. $10.2997$
Root an. cond. $3.20932$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 0.707i)2-s + (0.999 − 1.73i)4-s + (4.33 − 2.50i)5-s + (0.0413 − 6.99i)7-s − 2.82i·8-s + (3.54 − 6.13i)10-s + (−1.32 − 0.765i)11-s + 0.917·13-s + (−4.89 − 8.60i)14-s + (−2.00 − 3.46i)16-s + (−14.3 − 8.27i)17-s + (6.5 + 11.2i)19-s − 10.0i·20-s − 2.16·22-s + (10.3 − 5.98i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.867 − 0.500i)5-s + (0.00591 − 0.999i)7-s − 0.353i·8-s + (0.354 − 0.613i)10-s + (−0.120 − 0.0696i)11-s + 0.0705·13-s + (−0.349 − 0.614i)14-s + (−0.125 − 0.216i)16-s + (−0.843 − 0.486i)17-s + (0.342 + 0.592i)19-s − 0.500i·20-s − 0.0984·22-s + (0.450 − 0.260i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0692 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0692 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.0692 + 0.997i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ -0.0692 + 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.79168 - 1.92033i\)
\(L(\frac12)\) \(\approx\) \(1.79168 - 1.92033i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 \)
7 \( 1 + (-0.0413 + 6.99i)T \)
good5 \( 1 + (-4.33 + 2.50i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (1.32 + 0.765i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 0.917T + 169T^{2} \)
17 \( 1 + (14.3 + 8.27i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-6.5 - 11.2i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-10.3 + 5.98i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 33.5iT - 841T^{2} \)
31 \( 1 + (-8.66 + 15.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-27.8 - 48.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 6.53iT - 1.68e3T^{2} \)
43 \( 1 - 32.7T + 1.84e3T^{2} \)
47 \( 1 + (-46.3 + 26.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (72.4 + 41.8i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-41.4 - 23.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-40.2 - 69.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-1.21 + 2.09i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 83.3iT - 5.04e3T^{2} \)
73 \( 1 + (36.0 - 62.4i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-61.1 - 105. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 158. iT - 6.88e3T^{2} \)
89 \( 1 + (46.9 - 27.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 12.0T + 9.40e3T^{2} \)
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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

−10.96717118160330506252103937691, −10.02503922139000599270413250142, −9.390150544393762184876341214767, −8.088585495808072719798497654292, −6.91439462888629739237302916563, −5.93656733519971859515127026057, −4.89312900786708503216838322494, −3.92959311159191346596310904198, −2.43355821740430202116638724201, −1.00429830386208752243826456768, 2.07955701358562133481423551080, 3.07881078667561725164283439829, 4.65989878105538747419940356099, 5.70347410336296607592827779561, 6.38524355450549051709689445052, 7.40817316921128744274004027649, 8.705942415652203011052273203801, 9.408949829592562182250295244289, 10.64009158856005761905023703672, 11.38625265832642026961464057453

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