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Properties

Label 2-378-21.2-c2-0-0
Degree $2$
Conductor $378$
Sign $-0.893 - 0.449i$
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 + (3.11 + 1.79i)5-s + (−6.04 − 3.53i)7-s − 2.82i·8-s + (−2.54 − 4.40i)10-s + (−13.5 + 7.83i)11-s + 13.0·13-s + (4.89 + 8.60i)14-s + (−2.00 + 3.46i)16-s + (−22.9 + 13.2i)17-s + (6.5 − 11.2i)19-s + 7.18i·20-s + 22.1·22-s + (−17.8 − 10.2i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.622 + 0.359i)5-s + (−0.863 − 0.505i)7-s − 0.353i·8-s + (−0.254 − 0.440i)10-s + (−1.23 + 0.712i)11-s + 1.00·13-s + (0.349 + 0.614i)14-s + (−0.125 + 0.216i)16-s + (−1.34 + 0.778i)17-s + (0.342 − 0.592i)19-s + 0.359i·20-s + 1.00·22-s + (−0.774 − 0.447i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.893 - 0.449i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ -0.893 - 0.449i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0331731 + 0.139745i\)
\(L(\frac12)\) \(\approx\) \(0.0331731 + 0.139745i\)
\(L(2)\) 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 \( 1 + (6.04 + 3.53i)T \)
good5 \( 1 + (-3.11 - 1.79i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (13.5 - 7.83i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 13.0T + 169T^{2} \)
17 \( 1 + (22.9 - 13.2i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.5 + 11.2i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (17.8 + 10.2i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 9.48iT - 841T^{2} \)
31 \( 1 + (15.6 + 27.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (26.8 - 46.5i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 19.2iT - 1.68e3T^{2} \)
43 \( 1 + 76.7T + 1.84e3T^{2} \)
47 \( 1 + (-20.6 - 11.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (39.3 - 22.7i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (71.2 - 41.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (2.28 - 3.96i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-43.7 - 75.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 100. iT - 5.04e3T^{2} \)
73 \( 1 + (29.9 + 51.8i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-36.8 + 63.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 90.0iT - 6.88e3T^{2} \)
89 \( 1 + (109. + 63.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 0.0827T + 9.40e3T^{2} \)
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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

−11.20852305929741145062063446007, −10.41855769677750922186109823496, −9.955871760390825374042596452899, −8.904819774964423427436341233051, −7.914668082979105259859972322064, −6.79338646814985808909119071022, −6.08001399008992600216287713293, −4.45826103228241430539705827680, −3.10449272106525102221202156867, −1.93375269824777193735110132298, 0.07134758396239701447450004684, 1.98802159630345080740322983678, 3.39191598781618672069825421429, 5.27919586747687885304290418592, 5.91143436280194672958174125192, 6.91741351039156494583107826667, 8.167777869017238086401416373218, 8.934263352248787040769122564302, 9.673154463147019581240800498158, 10.59622579947631045386608294281

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