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Properties

Label 2-378-63.2-c2-0-4
Degree $2$
Conductor $378$
Sign $-0.645 - 0.764i$
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 + 9.69i·5-s + (6.99 + 0.244i)7-s + 2.82i·8-s + (−6.85 − 11.8i)10-s + 0.747i·11-s + (8.35 + 14.4i)13-s + (−8.74 + 4.64i)14-s + (−2.00 − 3.46i)16-s + (5.71 − 3.30i)17-s + (−0.429 + 0.744i)19-s + (16.7 + 9.69i)20-s + (−0.528 − 0.915i)22-s − 19.1i·23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + 1.93i·5-s + (0.999 + 0.0349i)7-s + 0.353i·8-s + (−0.685 − 1.18i)10-s + 0.0679i·11-s + (0.642 + 1.11i)13-s + (−0.624 + 0.331i)14-s + (−0.125 − 0.216i)16-s + (0.336 − 0.194i)17-s + (−0.0226 + 0.0391i)19-s + (0.839 + 0.484i)20-s + (−0.0240 − 0.0416i)22-s − 0.830i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 - 0.764i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.645 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.645 - 0.764i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ -0.645 - 0.764i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.547259 + 1.17809i\)
\(L(\frac12)\) \(\approx\) \(0.547259 + 1.17809i\)
\(L(2)\) 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 \( 1 + (-6.99 - 0.244i)T \)
good5 \( 1 - 9.69iT - 25T^{2} \)
11 \( 1 - 0.747iT - 121T^{2} \)
13 \( 1 + (-8.35 - 14.4i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-5.71 + 3.30i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (0.429 - 0.744i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + 19.1iT - 529T^{2} \)
29 \( 1 + (-19.2 - 11.1i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-19.6 + 34.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (34.2 - 59.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (27.4 - 15.8i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (30.3 - 52.6i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (1.03 - 0.597i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (20.6 - 11.9i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (19.6 + 11.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-4.17 - 7.22i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-8.09 + 14.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 36.6iT - 5.04e3T^{2} \)
73 \( 1 + (-1.75 - 3.03i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (50.1 + 86.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (21.5 + 12.4i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-53.5 - 30.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-22.1 + 38.4i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.35669789169376345486565208846, −10.52031820787044775057533578863, −9.833884366229871040665789044487, −8.537378408152083586486743732940, −7.70573488553566317568309388915, −6.74496952811999650354315568604, −6.18906444187143765896560660311, −4.58931812752820623596048324418, −3.08425682020928155878919466074, −1.80330901826338566375534622698, 0.75358600132657902177672540117, 1.72744450505946049350848301526, 3.71403638428132783298083819888, 4.94816718623076760188041442422, 5.67271188236253615302484913992, 7.48567442581773700637499839559, 8.494711620416904890563155963233, 8.606780849824742211643046303169, 9.847561622886132405966783557590, 10.76262571971265529133401186982

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