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Properties

Label 2-228-12.11-c1-0-27
Degree $2$
Conductor $228$
Sign $-0.329 + 0.944i$
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  + (−1.13 + 0.841i)2-s + (0.0680 − 1.73i)3-s + (0.583 − 1.91i)4-s − 1.36i·5-s + (1.37 + 2.02i)6-s − 1.12i·7-s + (0.946 + 2.66i)8-s + (−2.99 − 0.235i)9-s + (1.15 + 1.55i)10-s − 3.43·11-s + (−3.27 − 1.13i)12-s − 0.802·13-s + (0.950 + 1.28i)14-s + (−2.36 − 0.0930i)15-s + (−3.31 − 2.23i)16-s − 5.04i·17-s + ⋯
L(s)  = 1  + (−0.803 + 0.595i)2-s + (0.0392 − 0.999i)3-s + (0.291 − 0.956i)4-s − 0.611i·5-s + (0.563 + 0.826i)6-s − 0.426i·7-s + (0.334 + 0.942i)8-s + (−0.996 − 0.0784i)9-s + (0.364 + 0.491i)10-s − 1.03·11-s + (−0.944 − 0.329i)12-s − 0.222·13-s + (0.254 + 0.343i)14-s + (−0.611 − 0.0240i)15-s + (−0.829 − 0.558i)16-s − 1.22i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $-0.329 + 0.944i$
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: \((2,\ 228,\ (\ :1/2),\ -0.329 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.378382 - 0.532561i\)
\(L(\frac12)\) \(\approx\) \(0.378382 - 0.532561i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 - 0.841i)T \)
3 \( 1 + (-0.0680 + 1.73i)T \)
19 \( 1 - iT \)
good5 \( 1 + 1.36iT - 5T^{2} \)
7 \( 1 + 1.12iT - 7T^{2} \)
11 \( 1 + 3.43T + 11T^{2} \)
13 \( 1 + 0.802T + 13T^{2} \)
17 \( 1 + 5.04iT - 17T^{2} \)
23 \( 1 + 0.107T + 23T^{2} \)
29 \( 1 + 2.28iT - 29T^{2} \)
31 \( 1 + 6.32iT - 31T^{2} \)
37 \( 1 - 4.86T + 37T^{2} \)
41 \( 1 + 7.80iT - 41T^{2} \)
43 \( 1 - 4.40iT - 43T^{2} \)
47 \( 1 + 5.92T + 47T^{2} \)
53 \( 1 - 7.95iT - 53T^{2} \)
59 \( 1 - 14.9T + 59T^{2} \)
61 \( 1 + 11.9T + 61T^{2} \)
67 \( 1 - 8.03iT - 67T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 + 0.852T + 73T^{2} \)
79 \( 1 + 9.56iT - 79T^{2} \)
83 \( 1 - 6.54T + 83T^{2} \)
89 \( 1 + 4.58iT - 89T^{2} \)
97 \( 1 - 15.8T + 97T^{2} \)
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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

−11.87662385498401566099425443862, −10.92347297995573105232213568466, −9.773693763882164746772212330913, −8.770829802751794304468344985653, −7.80143474391649552503024178712, −7.21351392778846472557997277153, −5.95885625965512035214573085139, −4.92928110859036014470121151146, −2.44553573145140510258855274452, −0.67916512434894011780184259685, 2.50036762354526838628886003030, 3.54418352768096348626023436760, 5.03878835552149433253401145524, 6.53628046015443135135416569930, 7.956499062586749893334477136203, 8.740156271341420351565172172680, 9.816588596289697548630120268342, 10.56823375395670733698943677901, 11.11255319915670173021769076011, 12.25427122147308426424970513921

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