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Properties

Label 2-228-12.11-c1-0-22
Degree $2$
Conductor $228$
Sign $-0.0154 + 0.999i$
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  + (−0.557 − 1.29i)2-s + (1.17 + 1.27i)3-s + (−1.37 + 1.44i)4-s − 3.63i·5-s + (0.999 − 2.23i)6-s − 2.62i·7-s + (2.65 + 0.986i)8-s + (−0.237 + 2.99i)9-s + (−4.72 + 2.02i)10-s − 0.651·11-s + (−3.46 − 0.0534i)12-s + 2.40·13-s + (−3.41 + 1.46i)14-s + (4.62 − 4.26i)15-s + (−0.194 − 3.99i)16-s − 6.69i·17-s + ⋯
L(s)  = 1  + (−0.393 − 0.919i)2-s + (0.678 + 0.734i)3-s + (−0.689 + 0.724i)4-s − 1.62i·5-s + (0.408 − 0.912i)6-s − 0.993i·7-s + (0.937 + 0.348i)8-s + (−0.0793 + 0.996i)9-s + (−1.49 + 0.639i)10-s − 0.196·11-s + (−0.999 − 0.0154i)12-s + 0.667·13-s + (−0.912 + 0.391i)14-s + (1.19 − 1.10i)15-s + (−0.0485 − 0.998i)16-s − 1.62i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0154 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0154 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $-0.0154 + 0.999i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{228} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 228,\ (\ :1/2),\ -0.0154 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.812026 - 0.824649i\)
\(L(\frac12)\) \(\approx\) \(0.812026 - 0.824649i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.557 + 1.29i)T \)
3 \( 1 + (-1.17 - 1.27i)T \)
19 \( 1 - iT \)
good5 \( 1 + 3.63iT - 5T^{2} \)
7 \( 1 + 2.62iT - 7T^{2} \)
11 \( 1 + 0.651T + 11T^{2} \)
13 \( 1 - 2.40T + 13T^{2} \)
17 \( 1 + 6.69iT - 17T^{2} \)
23 \( 1 - 4.11T + 23T^{2} \)
29 \( 1 + 0.639iT - 29T^{2} \)
31 \( 1 - 9.21iT - 31T^{2} \)
37 \( 1 - 4.79T + 37T^{2} \)
41 \( 1 - 10.7iT - 41T^{2} \)
43 \( 1 - 1.99iT - 43T^{2} \)
47 \( 1 + 5.91T + 47T^{2} \)
53 \( 1 - 3.35iT - 53T^{2} \)
59 \( 1 + 3.51T + 59T^{2} \)
61 \( 1 - 3.79T + 61T^{2} \)
67 \( 1 - 11.0iT - 67T^{2} \)
71 \( 1 + 6.24T + 71T^{2} \)
73 \( 1 + 3.88T + 73T^{2} \)
79 \( 1 + 13.6iT - 79T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 - 2.37iT - 89T^{2} \)
97 \( 1 - 15.2T + 97T^{2} \)
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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

−11.85647521515441533857997204112, −10.89616005351884288560522346029, −9.895303892283546264343211065119, −9.157556346440964443278440535201, −8.438702882499054745125638022016, −7.49698330486657944102629063969, −5.01376040864055121875637962118, −4.42321094597526111749573273775, −3.13928164173524436663860974872, −1.16541733784491683311996506755, 2.19247446620385574408351066718, 3.65417358337376181824181179638, 5.87264887009614849202624214256, 6.46445208965844521717745747714, 7.46592560788719876346341285803, 8.340494693940344107960930829182, 9.243796425404653081671969391897, 10.40458761949714256391030147843, 11.34248613290534663790338379521, 12.77512332401449658431640410808

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