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Properties

Label 2-378-21.2-c2-0-17
Degree $2$
Conductor $378$
Sign $0.893 + 0.449i$
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 + (−3.11 − 1.79i)5-s + (−6.04 − 3.53i)7-s + 2.82i·8-s + (−2.54 − 4.40i)10-s + (13.5 − 7.83i)11-s + 13.0·13-s + (−4.89 − 8.60i)14-s + (−2.00 + 3.46i)16-s + (22.9 − 13.2i)17-s + (6.5 − 11.2i)19-s − 7.18i·20-s + 22.1·22-s + (17.8 + 10.2i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.622 − 0.359i)5-s + (−0.863 − 0.505i)7-s + 0.353i·8-s + (−0.254 − 0.440i)10-s + (1.23 − 0.712i)11-s + 1.00·13-s + (−0.349 − 0.614i)14-s + (−0.125 + 0.216i)16-s + (1.34 − 0.778i)17-s + (0.342 − 0.592i)19-s − 0.359i·20-s + 1.00·22-s + (0.774 + 0.447i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.893 + 0.449i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ 0.893 + 0.449i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.04917 - 0.486435i\)
\(L(\frac12)\) \(\approx\) \(2.04917 - 0.486435i\)
\(L(2)\) 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 \( 1 + (6.04 + 3.53i)T \)
good5 \( 1 + (3.11 + 1.79i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-13.5 + 7.83i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 13.0T + 169T^{2} \)
17 \( 1 + (-22.9 + 13.2i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.5 + 11.2i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-17.8 - 10.2i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 9.48iT - 841T^{2} \)
31 \( 1 + (15.6 + 27.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (26.8 - 46.5i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 19.2iT - 1.68e3T^{2} \)
43 \( 1 + 76.7T + 1.84e3T^{2} \)
47 \( 1 + (20.6 + 11.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-39.3 + 22.7i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-71.2 + 41.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (2.28 - 3.96i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-43.7 - 75.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 100. iT - 5.04e3T^{2} \)
73 \( 1 + (29.9 + 51.8i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-36.8 + 63.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 90.0iT - 6.88e3T^{2} \)
89 \( 1 + (-109. - 63.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 0.0827T + 9.40e3T^{2} \)
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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

−11.65111564519109398876073012128, −10.14973808786515042426029569297, −9.124597620861691632815571734888, −8.206654971862244244319025082371, −7.11174286379663895954080618911, −6.33581518703164475636628762634, −5.21995487792819041394612961478, −3.83813283233777663330516004608, −3.32105701881949198540935663928, −0.882932000678729013994040214464, 1.50690492784854959641075636471, 3.35414572877843118815927755152, 3.79211423479965401667675703031, 5.37914738248149109569545620571, 6.38345805012729053689044057252, 7.19190375867169698685573100091, 8.539765175495535462421672101125, 9.527972055348785371392144239177, 10.41916164267707458255722352019, 11.39364674746961105062079003749

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