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Properties

Label 2-378-21.11-c2-0-12
Degree $2$
Conductor $378$
Sign $0.744 + 0.667i$
Analytic cond. $10.2997$
Root an. cond. $3.20932$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 0.707i)2-s + (0.999 − 1.73i)4-s + (3.67 − 2.12i)5-s + (5.5 + 4.33i)7-s − 2.82i·8-s + (3 − 5.19i)10-s + 8·13-s + (9.79 + 1.41i)14-s + (−2.00 − 3.46i)16-s + (3.67 + 2.12i)17-s + (−2.5 − 4.33i)19-s − 8.48i·20-s + (18.3 − 10.6i)23-s + (−3.5 + 6.06i)25-s + (9.79 − 5.65i)26-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.734 − 0.424i)5-s + (0.785 + 0.618i)7-s − 0.353i·8-s + (0.300 − 0.519i)10-s + 0.615·13-s + (0.699 + 0.101i)14-s + (−0.125 − 0.216i)16-s + (0.216 + 0.124i)17-s + (−0.131 − 0.227i)19-s − 0.424i·20-s + (0.798 − 0.461i)23-s + (−0.140 + 0.242i)25-s + (0.376 − 0.217i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.744 + 0.667i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ 0.744 + 0.667i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.77292 - 1.06010i\)
\(L(\frac12)\) \(\approx\) \(2.77292 - 1.06010i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 \)
7 \( 1 + (-5.5 - 4.33i)T \)
good5 \( 1 + (-3.67 + 2.12i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (60.5 + 104. i)T^{2} \)
13 \( 1 - 8T + 169T^{2} \)
17 \( 1 + (-3.67 - 2.12i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-18.3 + 10.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 29.6iT - 841T^{2} \)
31 \( 1 + (14.5 - 25.1i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 72.1iT - 1.68e3T^{2} \)
43 \( 1 + 13T + 1.84e3T^{2} \)
47 \( 1 + (-18.3 + 10.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-55.1 - 31.8i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (29.3 + 16.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (20.5 + 35.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (64 - 110. i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 135. iT - 5.04e3T^{2} \)
73 \( 1 + (32.5 - 56.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 67.8iT - 6.88e3T^{2} \)
89 \( 1 + (91.8 - 53.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 103T + 9.40e3T^{2} \)
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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

−11.13730733883209050291580007783, −10.31354917017151447826769560167, −9.167489998583513299717666435323, −8.493770696180327677497794460152, −7.11516772471351959841727245094, −5.82622483071376582145143873264, −5.26235209838568270049473593736, −4.07509848269768761581082954615, −2.54358874566968616939508895100, −1.38193164750602679650848201736, 1.59958794753894537154487336450, 3.13210987436742973806680166714, 4.38418593464763985116505507572, 5.45006329713188517456984336402, 6.39978758381649550283941950198, 7.35669605797844917959634203588, 8.273761332616061949915077175609, 9.444079762041707014704476143987, 10.56409071652598231561337102687, 11.17966098487955085330561690098

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