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Properties

Label 2-378-9.7-c1-0-1
Degree $2$
Conductor $378$
Sign $0.173 - 0.984i$
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 + (1 + 1.73i)5-s + (0.5 − 0.866i)7-s + 0.999·8-s − 1.99·10-s + (0.5 − 0.866i)11-s + (3 + 5.19i)13-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 5·17-s − 7·19-s + (0.999 − 1.73i)20-s + (0.499 + 0.866i)22-s + (2 + 3.46i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.447 + 0.774i)5-s + (0.188 − 0.327i)7-s + 0.353·8-s − 0.632·10-s + (0.150 − 0.261i)11-s + (0.832 + 1.44i)13-s + (0.133 + 0.231i)14-s + (−0.125 + 0.216i)16-s + 1.21·17-s − 1.60·19-s + (0.223 − 0.387i)20-s + (0.106 + 0.184i)22-s + (0.417 + 0.722i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.943765 + 0.791912i\)
\(L(\frac12)\) \(\approx\) \(0.943765 + 0.791912i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3 - 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + (3.5 + 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8 + 13.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.21382476913412426416821379411, −10.66320997310013936672319637743, −9.649564808002223258409360373451, −8.786933477220466146940181912790, −7.79682463233054084338892125783, −6.64664905897367195457926409161, −6.22409520970669069439154576757, −4.79080207239947102306857522697, −3.47606948117361845475037224360, −1.67614299239604350579256902458, 1.07566152884837080457345135632, 2.58001998311368792871577853275, 4.01793267869358109412039514742, 5.25799444453769246158374559520, 6.19279108627659264670137224198, 7.82975644487966316227544025438, 8.478132158562124749100867162210, 9.361017597321515232368963967865, 10.28490712116099622030886667793, 11.01664605863384974468985906339

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