Properties

Label 2-378-27.22-c1-0-2
Degree $2$
Conductor $378$
Sign $-0.843 - 0.537i$
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  + (−0.766 + 0.642i)2-s + (−1.73 + 0.0419i)3-s + (0.173 − 0.984i)4-s + (−0.354 + 0.128i)5-s + (1.29 − 1.14i)6-s + (0.173 + 0.984i)7-s + (0.500 + 0.866i)8-s + (2.99 − 0.145i)9-s + (0.188 − 0.326i)10-s + (2.76 + 1.00i)11-s + (−0.259 + 1.71i)12-s + (−4.07 − 3.42i)13-s + (−0.766 − 0.642i)14-s + (0.608 − 0.238i)15-s + (−0.939 − 0.342i)16-s + (−3.02 + 5.24i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (−0.999 + 0.0241i)3-s + (0.0868 − 0.492i)4-s + (−0.158 + 0.0576i)5-s + (0.530 − 0.467i)6-s + (0.0656 + 0.372i)7-s + (0.176 + 0.306i)8-s + (0.998 − 0.0483i)9-s + (0.0596 − 0.103i)10-s + (0.832 + 0.303i)11-s + (−0.0748 + 0.494i)12-s + (−1.13 − 0.949i)13-s + (−0.204 − 0.171i)14-s + (0.156 − 0.0614i)15-s + (−0.234 − 0.0855i)16-s + (−0.734 + 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.843 - 0.537i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.843 - 0.537i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.113892 + 0.390652i\)
\(L(\frac12)\) \(\approx\) \(0.113892 + 0.390652i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 + (1.73 - 0.0419i)T \)
7 \( 1 + (-0.173 - 0.984i)T \)
good5 \( 1 + (0.354 - 0.128i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (-2.76 - 1.00i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (4.07 + 3.42i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (3.02 - 5.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0148 - 0.0256i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.37 - 7.81i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (4.78 - 4.01i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.0693 - 0.393i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-2.75 + 4.77i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.65 + 4.74i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.46 + 0.531i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.89 - 10.7i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 4.70T + 53T^{2} \)
59 \( 1 + (-6.22 + 2.26i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-2.33 - 13.2i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-9.08 - 7.61i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.81 + 4.87i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.51 + 9.55i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.32 - 2.79i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-8.38 + 7.03i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-1.32 - 2.28i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.37 + 3.04i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.60666794070204818929409545866, −10.80626523899257446734915998335, −9.875766699308422988218647111143, −9.133156604483244160115693043530, −7.78890717656917405926453929472, −7.08524414955153176464895839838, −5.96488086543851812581999998869, −5.25360315045918001520389789811, −3.90767189929148712599697483744, −1.71705050509711810793570997078, 0.36448191758729469063586101494, 2.15514647926568377148826382135, 4.06710492087380634572426944404, 4.86880034639831647028728542883, 6.49116624700842553566641773851, 7.03892055497106875535110667065, 8.250258228366100261260579618101, 9.502821290002570906342679323441, 10.00751458363818586599364975588, 11.23880976206829575049354671843

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