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Properties

Label 2-378-21.5-c1-0-7
Degree $2$
Conductor $378$
Sign $0.997 - 0.0633i$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.866 − 1.5i)5-s + (2 − 1.73i)7-s + 0.999i·8-s + (1.5 − 0.866i)10-s + (2.59 − 0.499i)14-s + (−0.5 + 0.866i)16-s + (−1.73 − 3i)17-s + (6 + 3.46i)19-s + 1.73·20-s + (−5.19 − 3i)23-s + (1 + 1.73i)25-s + (2.49 + 0.866i)28-s + 9i·29-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.387 − 0.670i)5-s + (0.755 − 0.654i)7-s + 0.353i·8-s + (0.474 − 0.273i)10-s + (0.694 − 0.133i)14-s + (−0.125 + 0.216i)16-s + (−0.420 − 0.727i)17-s + (1.37 + 0.794i)19-s + 0.387·20-s + (−1.08 − 0.625i)23-s + (0.200 + 0.346i)25-s + (0.472 + 0.163i)28-s + 1.67i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.997 - 0.0633i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.997 - 0.0633i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11940 + 0.0671812i\)
\(L(\frac12)\) \(\approx\) \(2.11940 + 0.0671812i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-2 + 1.73i)T \)
good5 \( 1 + (-0.866 + 1.5i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (1.73 + 3i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6 - 3.46i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 9iT - 29T^{2} \)
31 \( 1 + (-3 + 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (1.73 - 3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.59 - 1.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.06 + 10.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 - 1.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7 + 12.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6iT - 71T^{2} \)
73 \( 1 + (10.5 - 6.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 17.3T + 83T^{2} \)
89 \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−11.58581344562692100209151232561, −10.52984251731754702015359603858, −9.511942941367668005154788584297, −8.413153416934157637785576654959, −7.59090455383125247816478932374, −6.56359698330852523943630593997, −5.28318262389619172018236305331, −4.69961645524062533832628721809, −3.36073090217975545949937997347, −1.55177721551189855310407424136, 1.87964112982881891749086947556, 2.99044542318371344618484973509, 4.39005045929522300115716298579, 5.51232260619763605303286153358, 6.33363202206858744427286422153, 7.51552018699587945037862016351, 8.619447785236854367680274018152, 9.780871999835371775726955519410, 10.53121544419252486201399834854, 11.63958788032524503948841595735

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