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Properties

Label 2-228-19.11-c1-0-1
Degree $2$
Conductor $228$
Sign $0.942 - 0.334i$
Analytic cond. $1.82058$
Root an. cond. $1.34929$
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)3-s + (−1.82 + 3.15i)5-s + 4.64·7-s + (−0.499 − 0.866i)9-s + 0.354·11-s + (2.14 + 3.71i)13-s + (1.82 + 3.15i)15-s + (1.64 − 2.85i)17-s + (−2.64 − 3.46i)19-s + (2.32 − 4.02i)21-s + (2.82 + 4.88i)23-s + (−4.14 − 7.18i)25-s − 0.999·27-s + (−3.64 − 6.31i)29-s − 0.645·31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.815 + 1.41i)5-s + 1.75·7-s + (−0.166 − 0.288i)9-s + 0.106·11-s + (0.595 + 1.03i)13-s + (0.470 + 0.815i)15-s + (0.399 − 0.691i)17-s + (−0.606 − 0.794i)19-s + (0.506 − 0.877i)21-s + (0.588 + 1.01i)23-s + (−0.829 − 1.43i)25-s − 0.192·27-s + (−0.676 − 1.17i)29-s − 0.115·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $0.942 - 0.334i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{228} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 228,\ (\ :1/2),\ 0.942 - 0.334i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35409 + 0.233489i\)
\(L(\frac12)\) \(\approx\) \(1.35409 + 0.233489i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (2.64 + 3.46i)T \)
good5 \( 1 + (1.82 - 3.15i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 4.64T + 7T^{2} \)
11 \( 1 - 0.354T + 11T^{2} \)
13 \( 1 + (-2.14 - 3.71i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.64 + 2.85i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.82 - 4.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.64 + 6.31i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.645T + 31T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 + (2 - 3.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.67 - 4.63i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.64 + 4.58i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.46 + 6.00i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.82 - 6.62i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.96 + 12.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5 + 8.66i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.14 + 5.44i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.67 - 2.90i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.2T + 83T^{2} \)
89 \( 1 + (6.82 + 11.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.29 - 9.16i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.75942899585958860871184299746, −11.47626354465459274866594783738, −10.74900657025360383361924512229, −9.184482868962598575307530875937, −8.026063928137354569116804293347, −7.40523663416369012368474257661, −6.46349217305920116200753351129, −4.77679590566183089707330507766, −3.46621503037679029329843867293, −1.94265136438248305691524786488, 1.43737784354523566866754806779, 3.76585243242706886571605738005, 4.73906147432339918555538891198, 5.52250547127641139054474341285, 7.64618446506670680439236783675, 8.462202842434756412769772077754, 8.728984694046298523038727588929, 10.44152706274830947736625189681, 11.14285491817434855626676362325, 12.24532007020864631142059789702

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