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Properties

Label 2-228-57.56-c1-0-5
Degree $2$
Conductor $228$
Sign $0.606 + 0.794i$
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  + (1.58 − 0.707i)3-s − 2.23i·5-s − 7-s + (2.00 − 2.23i)9-s − 2.23i·11-s + 4.24i·13-s + (−1.58 − 3.53i)15-s − 2.23i·17-s + (1 + 4.24i)19-s + (−1.58 + 0.707i)21-s + 4.47i·23-s + (1.58 − 4.94i)27-s + 4.24i·31-s + (−1.58 − 3.53i)33-s + 2.23i·35-s + ⋯
L(s)  = 1  + (0.912 − 0.408i)3-s − 0.999i·5-s − 0.377·7-s + (0.666 − 0.745i)9-s − 0.674i·11-s + 1.17i·13-s + (−0.408 − 0.912i)15-s − 0.542i·17-s + (0.229 + 0.973i)19-s + (−0.345 + 0.154i)21-s + 0.932i·23-s + (0.304 − 0.952i)27-s + 0.762i·31-s + (−0.275 − 0.615i)33-s + 0.377i·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $0.606 + 0.794i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{228} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 228,\ (\ :1/2),\ 0.606 + 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40222 - 0.693667i\)
\(L(\frac12)\) \(\approx\) \(1.40222 - 0.693667i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.58 + 0.707i)T \)
19 \( 1 + (-1 - 4.24i)T \)
good5 \( 1 + 2.23iT - 5T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 + 2.23iT - 11T^{2} \)
13 \( 1 - 4.24iT - 13T^{2} \)
17 \( 1 + 2.23iT - 17T^{2} \)
23 \( 1 - 4.47iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 + 4.24iT - 37T^{2} \)
41 \( 1 - 9.48T + 41T^{2} \)
43 \( 1 + 7T + 43T^{2} \)
47 \( 1 - 11.1iT - 47T^{2} \)
53 \( 1 + 9.48T + 53T^{2} \)
59 \( 1 + 9.48T + 59T^{2} \)
61 \( 1 - 11T + 61T^{2} \)
67 \( 1 - 8.48iT - 67T^{2} \)
71 \( 1 + 9.48T + 71T^{2} \)
73 \( 1 - 5T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 8.94iT - 83T^{2} \)
89 \( 1 + 9.48T + 89T^{2} \)
97 \( 1 + 12.7iT - 97T^{2} \)
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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

−12.33090886343383339539003777289, −11.30967561064176197486831180209, −9.715537079226241580513690489009, −9.121930756973134315599662745561, −8.256413645644488743149093430888, −7.21172048854201565485589875521, −5.98239956767770817256545959499, −4.49012063105368663410121355418, −3.23107548140080737132588156357, −1.47855915060462285602673724242, 2.48930809001915653167932714819, 3.43449371691135284643896968006, 4.83087179585973810854288103502, 6.44685710099270129890573102673, 7.44375764756080496141996854807, 8.399486649778862251083002298731, 9.611498210121588089777678803151, 10.31518711999176003589813801455, 11.10259349654996059257444476463, 12.61290902232742326185644548774

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