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Properties

Label 2-378-7.4-c1-0-7
Degree $2$
Conductor $378$
Sign $0.968 + 0.250i$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1 − 1.73i)5-s + (0.5 − 2.59i)7-s − 0.999·8-s + (0.999 − 1.73i)10-s + (2.5 − 4.33i)11-s + 6·13-s + (2.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (−2 + 3.46i)17-s + (2 + 3.46i)19-s + 1.99·20-s + 5·22-s + (−2 − 3.46i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.447 − 0.774i)5-s + (0.188 − 0.981i)7-s − 0.353·8-s + (0.316 − 0.547i)10-s + (0.753 − 1.30i)11-s + 1.66·13-s + (0.668 − 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.485 + 0.840i)17-s + (0.458 + 0.794i)19-s + 0.447·20-s + 1.06·22-s + (−0.417 − 0.722i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53004 - 0.194979i\)
\(L(\frac12)\) \(\approx\) \(1.53004 - 0.194979i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-0.5 + 2.59i)T \)
good5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7T + 29T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + (6.5 - 11.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5T + 97T^{2} \)
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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

−11.23231644065076563021275437414, −10.70261601664224664085062975677, −9.071480386247876814458494402307, −8.475734715203401914282448175799, −7.67629939923188281731416952850, −6.39218317429906017235492956273, −5.67013401985455615137700811859, −4.06368059801025279216366444096, −3.77751641580697688276586904034, −1.07253666696515717695817810772, 1.84019554945691324312473519307, 3.16585513149263045912795950809, 4.22831925259862427095549030403, 5.48322240473819835554297886333, 6.58420840546510374209119620213, 7.55720783839625913117175217512, 8.980879058013000285634068643822, 9.488164441171811616303707442350, 10.83918973737132585995253991887, 11.45688228847364560918921616366

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