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Properties

Label 2-228-12.11-c1-0-23
Degree $2$
Conductor $228$
Sign $0.620 + 0.784i$
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  + (−0.221 + 1.39i)2-s + (−0.958 + 1.44i)3-s + (−1.90 − 0.619i)4-s − 3.67i·5-s + (−1.80 − 1.65i)6-s − 1.73i·7-s + (1.28 − 2.51i)8-s + (−1.16 − 2.76i)9-s + (5.13 + 0.814i)10-s − 5.81·11-s + (2.71 − 2.14i)12-s − 0.320·13-s + (2.41 + 0.383i)14-s + (5.29 + 3.52i)15-s + (3.23 + 2.35i)16-s − 1.14i·17-s + ⋯
L(s)  = 1  + (−0.156 + 0.987i)2-s + (−0.553 + 0.832i)3-s + (−0.950 − 0.309i)4-s − 1.64i·5-s + (−0.735 − 0.677i)6-s − 0.653i·7-s + (0.454 − 0.890i)8-s + (−0.387 − 0.922i)9-s + (1.62 + 0.257i)10-s − 1.75·11-s + (0.784 − 0.620i)12-s − 0.0888·13-s + (0.645 + 0.102i)14-s + (1.36 + 0.909i)15-s + (0.808 + 0.588i)16-s − 0.278i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $0.620 + 0.784i$
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: \((2,\ 228,\ (\ :1/2),\ 0.620 + 0.784i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.460369 - 0.222805i\)
\(L(\frac12)\) \(\approx\) \(0.460369 - 0.222805i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.221 - 1.39i)T \)
3 \( 1 + (0.958 - 1.44i)T \)
19 \( 1 + iT \)
good5 \( 1 + 3.67iT - 5T^{2} \)
7 \( 1 + 1.73iT - 7T^{2} \)
11 \( 1 + 5.81T + 11T^{2} \)
13 \( 1 + 0.320T + 13T^{2} \)
17 \( 1 + 1.14iT - 17T^{2} \)
23 \( 1 - 2.06T + 23T^{2} \)
29 \( 1 - 7.38iT - 29T^{2} \)
31 \( 1 + 6.00iT - 31T^{2} \)
37 \( 1 + 5.56T + 37T^{2} \)
41 \( 1 + 4.66iT - 41T^{2} \)
43 \( 1 + 11.9iT - 43T^{2} \)
47 \( 1 + 2.44T + 47T^{2} \)
53 \( 1 + 3.33iT - 53T^{2} \)
59 \( 1 + 3.98T + 59T^{2} \)
61 \( 1 - 8.88T + 61T^{2} \)
67 \( 1 - 6.97iT - 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 - 3.33T + 73T^{2} \)
79 \( 1 - 4.89iT - 79T^{2} \)
83 \( 1 + 1.65T + 83T^{2} \)
89 \( 1 - 8.78iT - 89T^{2} \)
97 \( 1 - 3.03T + 97T^{2} \)
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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

−12.34735104924540302833326157381, −10.81653739840859571346989854811, −9.987837710551630156152752109545, −9.033302169831737218899680546239, −8.248929626551226208038061331308, −7.07949661762388611804341248326, −5.32387530680149533411864766752, −5.19401549615017832920028560447, −3.98391626239870772591308694908, −0.46426675983916166612926754872, 2.25851083644917550795066190411, 3.04317257426280054064701165926, 5.09728105087394857749872835177, 6.22037198073950108274734266000, 7.50591575635203153982751158685, 8.254945920936154259886273987386, 9.923565018224605150409425397592, 10.69642755312586029150704444659, 11.26252819200483435651429828508, 12.23198883884648023713230901908

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