↑

Properties

Label 2-378-63.58-c1-0-2
Degree $2$
Conductor $378$
Sign $0.580 - 0.814i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + (1.5 + 2.59i)5-s + (−2 + 1.73i)7-s + 8-s + (1.5 + 2.59i)10-s + (−1.5 + 2.59i)11-s + (0.5 − 0.866i)13-s + (−2 + 1.73i)14-s + 16-s + (1.5 + 2.59i)17-s + (3.5 − 6.06i)19-s + (1.5 + 2.59i)20-s + (−1.5 + 2.59i)22-s + (−4.5 − 7.79i)23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.670 + 1.16i)5-s + (−0.755 + 0.654i)7-s + 0.353·8-s + (0.474 + 0.821i)10-s + (−0.452 + 0.783i)11-s + (0.138 − 0.240i)13-s + (−0.534 + 0.462i)14-s + 0.250·16-s + (0.363 + 0.630i)17-s + (0.802 − 1.39i)19-s + (0.335 + 0.580i)20-s + (−0.319 + 0.553i)22-s + (−0.938 − 1.62i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.580 - 0.814i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.580 - 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85110 + 0.953883i\)
\(L(\frac12)\) \(\approx\) \(1.85110 + 0.953883i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 + (2 - 1.73i)T \)
good5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.5 + 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.63142723850638055649090829029, −10.36785429474014655716130418144, −10.12129034070783833462590717176, −8.805620007560151480059951310191, −7.41660624080009576011790407789, −6.51363568889788329378031811003, −5.88717189604746363667709496205, −4.62300575707966355902197951268, −3.04840906576683980501656936420, −2.39765642057837759521115600740, 1.27850408908858898470121758468, 3.11563751589217092738468726414, 4.24600647409874965225879461691, 5.50504203684900015512796753427, 6.02736319945167863271170682712, 7.42306523483932211464784717123, 8.376924833528792729177956035729, 9.689956666291715631542457653511, 10.07696488686269583473884902051, 11.51312794726508477628680139939

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