Properties

Label 2-378-27.7-c1-0-5
Degree $2$
Conductor $378$
Sign $0.266 - 0.963i$
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 + (0.0792 + 1.73i)3-s + (0.766 − 0.642i)4-s + (0.488 + 2.76i)5-s + (0.666 + 1.59i)6-s + (−0.766 − 0.642i)7-s + (0.500 − 0.866i)8-s + (−2.98 + 0.274i)9-s + (1.40 + 2.43i)10-s + (−0.288 + 1.63i)11-s + (1.17 + 1.27i)12-s + (3.64 + 1.32i)13-s + (−0.939 − 0.342i)14-s + (−4.75 + 1.06i)15-s + (0.173 − 0.984i)16-s + (−0.595 − 1.03i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (0.0457 + 0.998i)3-s + (0.383 − 0.321i)4-s + (0.218 + 1.23i)5-s + (0.271 + 0.652i)6-s + (−0.289 − 0.242i)7-s + (0.176 − 0.306i)8-s + (−0.995 + 0.0913i)9-s + (0.444 + 0.770i)10-s + (−0.0868 + 0.492i)11-s + (0.338 + 0.367i)12-s + (1.01 + 0.367i)13-s + (−0.251 − 0.0914i)14-s + (−1.22 + 0.274i)15-s + (0.0434 − 0.246i)16-s + (−0.144 − 0.250i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.266 - 0.963i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55741 + 1.18493i\)
\(L(\frac12)\) \(\approx\) \(1.55741 + 1.18493i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0792 - 1.73i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
good5 \( 1 + (-0.488 - 2.76i)T + (-4.69 + 1.71i)T^{2} \)
11 \( 1 + (0.288 - 1.63i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-3.64 - 1.32i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.595 + 1.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.258 + 0.447i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.18 - 2.67i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-0.347 + 0.126i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-3.66 + 3.07i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (2.22 + 3.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-10.7 - 3.90i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.20 + 6.80i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-6.16 - 5.17i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 + (2.21 + 12.5i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (9.64 + 8.09i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (2.00 + 0.729i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (6.05 + 10.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.52 + 4.37i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.7 - 3.90i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (3.38 - 1.23i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (7.71 - 13.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.35 - 7.69i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.25020479457801660671226255709, −10.77401616180027924910457174282, −9.971144359254958038172307515568, −9.145192755101485049583484717076, −7.66176123575777092023751449581, −6.51666111588094013606313612028, −5.74888785077206386028318170075, −4.36507824300925617205426780330, −3.51446706277577436564172761299, −2.44217584295336692521440529001, 1.18624494075809294406501811827, 2.78759128671033425122110675635, 4.25365732703611287085346161431, 5.65534442678552129116364397591, 6.06534317618298093053801438048, 7.33888297745117630486920334794, 8.520729537247766352778643767442, 8.778896650702308954971648937912, 10.41674371088576559236877961511, 11.59242069658619848086976004548

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