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Properties

Label 2-228-57.17-c2-0-6
Degree $2$
Conductor $228$
Sign $0.885 + 0.465i$
Analytic cond. $6.21255$
Root an. cond. $2.49249$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.29 − 1.92i)3-s + (3.68 + 6.38i)7-s + (1.56 + 8.86i)9-s + (19.4 − 16.3i)13-s + (5.5 − 18.1i)19-s + (3.84 − 21.7i)21-s + (19.1 − 16.0i)25-s + (13.4 − 23.3i)27-s + (24.4 + 42.3i)31-s + 72.7·37-s − 76.2·39-s + (−73.3 + 26.6i)43-s + (−2.69 + 4.66i)49-s + (−47.7 + 31.1i)57-s + (34.1 + 12.4i)61-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)3-s + (0.526 + 0.912i)7-s + (0.173 + 0.984i)9-s + (1.49 − 1.25i)13-s + (0.289 − 0.957i)19-s + (0.182 − 1.03i)21-s + (0.766 − 0.642i)25-s + (0.499 − 0.866i)27-s + (0.789 + 1.36i)31-s + 1.96·37-s − 1.95·39-s + (−1.70 + 0.620i)43-s + (−0.0549 + 0.0951i)49-s + (−0.837 + 0.547i)57-s + (0.560 + 0.203i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.465i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.885 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $0.885 + 0.465i$
Analytic conductor: \(6.21255\)
Root analytic conductor: \(2.49249\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{228} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 228,\ (\ :1),\ 0.885 + 0.465i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.30804 - 0.322884i\)
\(L(\frac12)\) \(\approx\) \(1.30804 - 0.322884i\)
\(L(2)\) 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 \( 1 + (2.29 + 1.92i)T \)
19 \( 1 + (-5.5 + 18.1i)T \)
good5 \( 1 + (-19.1 + 16.0i)T^{2} \)
7 \( 1 + (-3.68 - 6.38i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-19.4 + 16.3i)T + (29.3 - 166. i)T^{2} \)
17 \( 1 + (271. + 98.8i)T^{2} \)
23 \( 1 + (-405. - 340. i)T^{2} \)
29 \( 1 + (790. - 287. i)T^{2} \)
31 \( 1 + (-24.4 - 42.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 72.7T + 1.36e3T^{2} \)
41 \( 1 + (-291. - 1.65e3i)T^{2} \)
43 \( 1 + (73.3 - 26.6i)T + (1.41e3 - 1.18e3i)T^{2} \)
47 \( 1 + (2.07e3 - 755. i)T^{2} \)
53 \( 1 + (-2.15e3 - 1.80e3i)T^{2} \)
59 \( 1 + (3.27e3 + 1.19e3i)T^{2} \)
61 \( 1 + (-34.1 - 12.4i)T + (2.85e3 + 2.39e3i)T^{2} \)
67 \( 1 + (-10.0 - 56.9i)T + (-4.21e3 + 1.53e3i)T^{2} \)
71 \( 1 + (-3.86e3 + 3.24e3i)T^{2} \)
73 \( 1 + (3.19 + 2.67i)T + (925. + 5.24e3i)T^{2} \)
79 \( 1 + (117. + 98.5i)T + (1.08e3 + 6.14e3i)T^{2} \)
83 \( 1 + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-1.37e3 + 7.80e3i)T^{2} \)
97 \( 1 + (29.3 - 166. i)T + (-8.84e3 - 3.21e3i)T^{2} \)
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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

−11.82614882649893493441941485890, −11.15831237932765090005586959913, −10.26111165791513674207713139692, −8.686257156998327463190273878605, −8.041771032607516608015772363358, −6.66927265476503191124130029692, −5.76028009424723331432559199265, −4.82291007713607630881146293561, −2.82061246728182215626196206245, −1.09146943938086855276090067051, 1.21950226075927087100220019127, 3.73758565371178531809480206270, 4.51963885814942476092872264115, 5.88937631779589853312109356332, 6.82342445146066613250013031622, 8.145120685682907246096709792150, 9.328568818827825970026282677860, 10.27151176690066127610310764926, 11.23681697510629158039038134119, 11.62622675350760218192973483434

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