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Properties

Label 2-378-9.5-c2-0-5
Degree $2$
Conductor $378$
Sign $0.976 + 0.216i$
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 + (4.50 + 2.60i)5-s + (1.32 + 2.29i)7-s − 2.82i·8-s + 7.36·10-s + (11.4 − 6.62i)11-s + (−4.01 + 6.95i)13-s + (3.24 + 1.87i)14-s + (−2.00 − 3.46i)16-s + 8.82i·17-s − 3.48·19-s + (9.01 − 5.20i)20-s + (9.36 − 16.2i)22-s + (32.7 + 18.9i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.901 + 0.520i)5-s + (0.188 + 0.327i)7-s − 0.353i·8-s + 0.736·10-s + (1.04 − 0.602i)11-s + (−0.308 + 0.534i)13-s + (0.231 + 0.133i)14-s + (−0.125 − 0.216i)16-s + 0.519i·17-s − 0.183·19-s + (0.450 − 0.260i)20-s + (0.425 − 0.737i)22-s + (1.42 + 0.822i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.976 + 0.216i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ 0.976 + 0.216i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.92969 - 0.320350i\)
\(L(\frac12)\) \(\approx\) \(2.92969 - 0.320350i\)
\(L(2)\) 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 \( 1 + (-1.32 - 2.29i)T \)
good5 \( 1 + (-4.50 - 2.60i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-11.4 + 6.62i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (4.01 - 6.95i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 8.82iT - 289T^{2} \)
19 \( 1 + 3.48T + 361T^{2} \)
23 \( 1 + (-32.7 - 18.9i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-13.4 + 7.76i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-11.9 + 20.6i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 19.2T + 1.36e3T^{2} \)
41 \( 1 + (29.2 + 16.8i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (22.7 + 39.4i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (52.1 - 30.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 63.7iT - 2.80e3T^{2} \)
59 \( 1 + (-94.5 - 54.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (39.6 + 68.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (53.3 - 92.4i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 94.1iT - 5.04e3T^{2} \)
73 \( 1 + 3.40T + 5.32e3T^{2} \)
79 \( 1 + (78.0 + 135. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (130. - 75.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 29.5iT - 7.92e3T^{2} \)
97 \( 1 + (-40.7 - 70.6i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.33826314122208808716595444790, −10.25549401177869173383006189209, −9.473779512383530946053846065595, −8.523567687380578558556358857208, −6.95403690278456105523071182558, −6.23410866318372050218003024897, −5.30271573829163113646130283024, −4.00197356697636290467528488528, −2.75574012323248443025038808879, −1.53373169497038232594670447548, 1.39713350556637753091321353521, 2.96131246247620967724573097828, 4.50013896008952933366439265029, 5.18004033972169259208800586093, 6.39932634274741961771046581649, 7.13749747724133243252244468580, 8.408125152213111504441507204113, 9.327511201779658877901309396152, 10.19247048595059344875363600269, 11.32855538088438436195312140555

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