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Properties

Label 2-378-63.4-c1-0-5
Degree $2$
Conductor $378$
Sign $0.999 + 0.00294i$
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 + 3.18·5-s + (0.710 − 2.54i)7-s + 0.999·8-s + (−1.59 + 2.75i)10-s − 3.18·11-s + (2.85 − 4.93i)13-s + (1.85 + 1.88i)14-s + (−0.5 + 0.866i)16-s + (0.760 − 1.31i)17-s + (−0.641 − 1.11i)19-s + (−1.59 − 2.75i)20-s + (1.59 − 2.75i)22-s − 2.23·23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + 1.42·5-s + (0.268 − 0.963i)7-s + 0.353·8-s + (−0.503 + 0.871i)10-s − 0.959·11-s + (0.790 − 1.36i)13-s + (0.494 + 0.505i)14-s + (−0.125 + 0.216i)16-s + (0.184 − 0.319i)17-s + (−0.147 − 0.254i)19-s + (−0.355 − 0.616i)20-s + (0.339 − 0.587i)22-s − 0.466·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.999 + 0.00294i$
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: \((2,\ 378,\ (\ :1/2),\ 0.999 + 0.00294i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38973 - 0.00204684i\)
\(L(\frac12)\) \(\approx\) \(1.38973 - 0.00204684i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.710 + 2.54i)T \)
good5 \( 1 - 3.18T + 5T^{2} \)
11 \( 1 + 3.18T + 11T^{2} \)
13 \( 1 + (-2.85 + 4.93i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.760 + 1.31i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.641 + 1.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.23T + 23T^{2} \)
29 \( 1 + (-3.54 - 6.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.71 - 8.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.80 + 4.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.41 - 5.91i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.91 - 5.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.02 - 1.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.562 + 0.974i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.56 - 2.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.48 + 9.49i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.69T + 71T^{2} \)
73 \( 1 + (2.48 - 4.30i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.06 + 3.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.03 - 6.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.112 + 0.195i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.42 - 12.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.65854808284521445423340039810, −10.56567793432145524084938723653, −9.603360608405160181028994887321, −8.485915307753444226217294737256, −7.66682594770704328895198652806, −6.56306645923745034013988641708, −5.65171965761350440612036963254, −4.81146958152562143543122396892, −2.97002455780179579234501584556, −1.21296200663693563952143453751, 1.80486715827586510244283793887, 2.58332120179096718550293936343, 4.35302998245710172781996760259, 5.67587743052420171466070092200, 6.33494758858488591759251296799, 7.959176723345529284096229554916, 8.817130951228127687827628812753, 9.634404470272099906715239586508, 10.28982942262321705074592652744, 11.35899864675022016392162119692

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