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Properties

Label 2-378-9.4-c1-0-1
Degree $2$
Conductor $378$
Sign $0.918 - 0.394i$
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 + (−0.686 + 1.18i)5-s + (−0.5 − 0.866i)7-s + 0.999·8-s + 1.37·10-s + (2.18 + 3.78i)11-s + (−1 + 1.73i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + 4.37·17-s + 5·19-s + (−0.686 − 1.18i)20-s + (2.18 − 3.78i)22-s + (−3.68 + 6.38i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.306 + 0.531i)5-s + (−0.188 − 0.327i)7-s + 0.353·8-s + 0.433·10-s + (0.659 + 1.14i)11-s + (−0.277 + 0.480i)13-s + (−0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s + 1.06·17-s + 1.14·19-s + (−0.153 − 0.265i)20-s + (0.466 − 0.807i)22-s + (−0.768 + 1.33i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.918 - 0.394i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.918 - 0.394i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.996713 + 0.204789i\)
\(L(\frac12)\) \(\approx\) \(0.996713 + 0.204789i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.686 - 1.18i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.18 - 3.78i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.37T + 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + (3.68 - 6.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.37 - 2.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-5.18 + 8.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.55 + 7.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.74T + 53T^{2} \)
59 \( 1 + (-3.55 + 6.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.05 - 12.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.55 - 13.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 + 5.11T + 73T^{2} \)
79 \( 1 + (6.05 + 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.74 + 4.75i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.25T + 89T^{2} \)
97 \( 1 + (4.55 + 7.89i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.63723860492321696046302496073, −10.33639435740983412785178352322, −9.792344904414724789176825382712, −8.911771334827100226404133927301, −7.37209212560432781948332279121, −7.22057338555977952145283121708, −5.51855206258890072817998252812, −4.12592636405168422921552723852, −3.19346068723694222992876887276, −1.57879843614538794365131137200, 0.866087841543724400601761598973, 3.08583661968968612764015978118, 4.51131880856165925700181116171, 5.68280733791041058273165830676, 6.43584126441933528374313608425, 7.84261529258016309172135468118, 8.342269316485315092441089560446, 9.374926029770812745189207620591, 10.13793023661645728921978327955, 11.36455164675024265570475509744

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