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Properties

Label 2-378-63.31-c2-0-10
Degree $2$
Conductor $378$
Sign $-0.618 + 0.785i$
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  + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s − 7.53i·5-s + (6.72 + 1.95i)7-s − 2.82·8-s + (−9.23 − 5.32i)10-s + 20.8·11-s + (−10.7 − 6.18i)13-s + (7.14 − 6.84i)14-s + (−2.00 + 3.46i)16-s + (−15.2 − 8.80i)17-s + (5.57 − 3.21i)19-s + (−13.0 + 7.53i)20-s + (14.7 − 25.5i)22-s + 1.49·23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s − 1.50i·5-s + (0.960 + 0.279i)7-s − 0.353·8-s + (−0.923 − 0.532i)10-s + 1.89·11-s + (−0.824 − 0.475i)13-s + (0.510 − 0.489i)14-s + (−0.125 + 0.216i)16-s + (−0.897 − 0.518i)17-s + (0.293 − 0.169i)19-s + (−0.652 + 0.376i)20-s + (0.671 − 1.16i)22-s + 0.0648·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.618 + 0.785i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ -0.618 + 0.785i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.927493 - 1.91034i\)
\(L(\frac12)\) \(\approx\) \(0.927493 - 1.91034i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 \)
7 \( 1 + (-6.72 - 1.95i)T \)
good5 \( 1 + 7.53iT - 25T^{2} \)
11 \( 1 - 20.8T + 121T^{2} \)
13 \( 1 + (10.7 + 6.18i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (15.2 + 8.80i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-5.57 + 3.21i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 - 1.49T + 529T^{2} \)
29 \( 1 + (11.8 + 20.5i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (27.7 - 15.9i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-10.3 - 18.0i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (51.7 + 29.8i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-1.17 - 2.04i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-50.7 - 29.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-33.3 + 57.7i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-38.5 + 22.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-11.3 - 6.53i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-16.7 - 28.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 15.9T + 5.04e3T^{2} \)
73 \( 1 + (-96.5 - 55.7i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (26.1 - 45.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (21.1 - 12.2i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (62.5 - 36.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-12.1 + 7.00i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−11.18761522163272445012560166503, −9.734326008626804904496651132555, −9.043463715447490385532574595596, −8.394141798385406912246931283287, −7.00129093828830144286754830554, −5.53191411468392175233477714656, −4.78636938984785096749569077595, −3.93970794509006203905466652098, −2.04151474870166390247512809489, −0.907719459972892169108152254985, 1.97659938577225498295165070436, 3.58698743890326488658156235182, 4.44671373961167284215004587922, 5.90146517289617649992796450860, 6.94048052994058560965445366785, 7.25470992607612863595803954546, 8.609385507614109296262984536846, 9.563488251694218708020726041132, 10.77064134493309145415085750775, 11.45109312700020015008859776567

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