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Properties

Label 2-228-12.11-c1-0-24
Degree $2$
Conductor $228$
Sign $0.963 + 0.268i$
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 + (2.22 + 1.01i)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 + (−0.929 + 3.33i)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 + (0.909 + 0.416i)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 + (−0.268 + 0.963i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $0.963 + 0.268i$
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: \((2,\ 228,\ (\ :1/2),\ 0.963 + 0.268i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42920 - 0.195383i\)
\(L(\frac12)\) \(\approx\) \(1.42920 - 0.195383i\)
\(L(\frac{3}{2})\) 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

−12.29209591391003131204499963374, −11.98116893028802541011043515733, −9.497372179337380837275081438514, −8.873135747374596451418934642323, −8.420615860233018850275074058612, −7.14230713842623159474434425420, −6.11968676006249319085385483899, −5.02040661520862565400419377539, −3.72424927887596454800658966457, −1.31582774282553976503663384065, 2.30340842372381552734900607726, 3.62656181170971184509657365444, 4.14902596879585488249070219425, 6.01373875306534391499877206798, 7.30130654755911140855292379379, 8.684567807200179044121621886387, 9.784405350302589900592338909830, 10.29664111441949456518526018195, 11.21997029495385624942350156045, 11.82049486186858785449949091464

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