Properties

Label 2-378-21.5-c1-0-1
Degree $2$
Conductor $378$
Sign $0.444 - 0.895i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (0.5 + 2.59i)7-s − 0.999i·8-s + (−2.59 + 1.5i)11-s + 3.46i·13-s + (0.866 − 2.5i)14-s + (−0.5 + 0.866i)16-s + 3·22-s + (2.5 + 4.33i)25-s + (1.73 − 2.99i)26-s + (−2 + 1.73i)28-s + 9i·29-s + (−1.5 + 0.866i)31-s + (0.866 − 0.499i)32-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.188 + 0.981i)7-s − 0.353i·8-s + (−0.783 + 0.452i)11-s + 0.960i·13-s + (0.231 − 0.668i)14-s + (−0.125 + 0.216i)16-s + 0.639·22-s + (0.5 + 0.866i)25-s + (0.339 − 0.588i)26-s + (−0.377 + 0.327i)28-s + 1.67i·29-s + (−0.269 + 0.155i)31-s + (0.153 − 0.0883i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.444 - 0.895i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.444 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.736912 + 0.457182i\)
\(L(\frac12)\) \(\approx\) \(0.736912 + 0.457182i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 9iT - 29T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (5.19 - 9i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.19 + 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.59 + 4.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (12 + 6.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + (-4.5 + 2.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.19T + 83T^{2} \)
89 \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.66iT - 97T^{2} \)
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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

−11.29954258538539719251393088713, −10.76714711043957731812398663717, −9.439396007299702198485719503798, −9.035563811733235647825650502911, −7.909285770097460285166737956836, −7.01071852508844524120473229476, −5.74087962421612464081947245288, −4.59758807589712281583841461350, −3.00258585574345978194414925958, −1.82160538634149474370599481218, 0.71997311433045778388463728406, 2.70612079438598448175909728142, 4.27046357432050017177124117155, 5.52192614860960498756767291182, 6.53875476603521524901567443317, 7.77841752412998338198691453115, 8.086574231579406264468996357959, 9.415350542897202504801765229083, 10.39501869746702445076413515214, 10.80953530341165412227174786886

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