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Properties

Label 2-378-63.4-c1-0-1
Degree $2$
Conductor $378$
Sign $-0.609 - 0.792i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s − 0.593·5-s + (−0.0665 + 2.64i)7-s + 0.999·8-s + (0.296 − 0.514i)10-s + 0.593·11-s + (−1.25 + 2.17i)13-s + (−2.25 − 1.38i)14-s + (−0.5 + 0.866i)16-s + (−1.46 + 2.52i)17-s + (2.69 + 4.66i)19-s + (0.296 + 0.514i)20-s + (−0.296 + 0.514i)22-s − 4.46·23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.265·5-s + (−0.0251 + 0.999i)7-s + 0.353·8-s + (0.0938 − 0.162i)10-s + 0.178·11-s + (−0.348 + 0.603i)13-s + (−0.603 − 0.368i)14-s + (−0.125 + 0.216i)16-s + (−0.354 + 0.613i)17-s + (0.617 + 1.06i)19-s + (0.0663 + 0.114i)20-s + (−0.0632 + 0.109i)22-s − 0.930·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.609 - 0.792i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.609 - 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.376658 + 0.764442i\)
\(L(\frac12)\) \(\approx\) \(0.376658 + 0.764442i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (0.0665 - 2.64i)T \)
good5 \( 1 + 0.593T + 5T^{2} \)
11 \( 1 - 0.593T + 11T^{2} \)
13 \( 1 + (1.25 - 2.17i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.46 - 2.52i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.69 - 4.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.46T + 23T^{2} \)
29 \( 1 + (-3.09 - 5.36i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.93 - 6.81i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.136 + 0.236i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.58 + 9.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.08 + 10.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.02 - 6.97i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.32 - 7.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.32 + 5.75i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.956 - 1.65i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 + (-3.95 + 6.85i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.62 + 8.00i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.85 + 6.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.21 - 10.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.86 - 10.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.98647974666041346932303693560, −10.57594986926823079797590206614, −9.727104775925431944125837671449, −8.755022131637625542872844524264, −8.113658215841622770996367541887, −6.95522896531781704814822976055, −6.02304779982039994760018645355, −5.06043877812093635130566172818, −3.69721634376078247122664610306, −1.92734219114311631730865400329, 0.64848224129327926610869568510, 2.55147765920002646137517570120, 3.86472451637733685311821579134, 4.81664585564264533858711100717, 6.39360013178579474315822769517, 7.56037032028413341592125498858, 8.135812797060768007880511251695, 9.597382888290522271631012122468, 9.956711435928012677571310782686, 11.21847873660925762458660780044

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