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Properties

Label 2-228-12.11-c1-0-28
Degree $2$
Conductor $228$
Sign $0.726 + 0.687i$
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  + (1.08 − 0.901i)2-s + (1.45 + 0.932i)3-s + (0.375 − 1.96i)4-s − 2.01i·5-s + (2.43 − 0.298i)6-s + 2.54i·7-s + (−1.36 − 2.47i)8-s + (1.25 + 2.72i)9-s + (−1.81 − 2.19i)10-s + 0.727·11-s + (2.38 − 2.51i)12-s − 6.54·13-s + (2.28 + 2.76i)14-s + (1.88 − 2.94i)15-s + (−3.71 − 1.47i)16-s + 1.39i·17-s + ⋯
L(s)  = 1  + (0.770 − 0.637i)2-s + (0.842 + 0.538i)3-s + (0.187 − 0.982i)4-s − 0.902i·5-s + (0.992 − 0.121i)6-s + 0.960i·7-s + (−0.481 − 0.876i)8-s + (0.419 + 0.907i)9-s + (−0.574 − 0.695i)10-s + 0.219·11-s + (0.687 − 0.726i)12-s − 1.81·13-s + (0.611 + 0.740i)14-s + (0.485 − 0.760i)15-s + (−0.929 − 0.369i)16-s + 0.338i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(228\)    =    \(2^{2} \cdot 3 \cdot 19\)
Sign: $0.726 + 0.687i$
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: \((2,\ 228,\ (\ :1/2),\ 0.726 + 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03523 - 0.810275i\)
\(L(\frac12)\) \(\approx\) \(2.03523 - 0.810275i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.08 + 0.901i)T \)
3 \( 1 + (-1.45 - 0.932i)T \)
19 \( 1 + iT \)
good5 \( 1 + 2.01iT - 5T^{2} \)
7 \( 1 - 2.54iT - 7T^{2} \)
11 \( 1 - 0.727T + 11T^{2} \)
13 \( 1 + 6.54T + 13T^{2} \)
17 \( 1 - 1.39iT - 17T^{2} \)
23 \( 1 - 5.80T + 23T^{2} \)
29 \( 1 - 2.06iT - 29T^{2} \)
31 \( 1 - 1.55iT - 31T^{2} \)
37 \( 1 + 9.59T + 37T^{2} \)
41 \( 1 + 7.32iT - 41T^{2} \)
43 \( 1 - 7.93iT - 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 + 9.29iT - 53T^{2} \)
59 \( 1 + 1.87T + 59T^{2} \)
61 \( 1 - 3.98T + 61T^{2} \)
67 \( 1 + 5.40iT - 67T^{2} \)
71 \( 1 - 14.6T + 71T^{2} \)
73 \( 1 - 5.82T + 73T^{2} \)
79 \( 1 - 4.95iT - 79T^{2} \)
83 \( 1 + 1.92T + 83T^{2} \)
89 \( 1 + 9.00iT - 89T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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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

−12.48521995038311058997091176086, −11.28107303833004710973872511690, −10.06225058352114838948121620084, −9.294337476310044634412105227845, −8.556597581975296434726421154435, −7.00245171250545195924523903152, −5.19865898494455124308165823367, −4.80672116542128959738126451402, −3.24703815113364925264037267670, −2.05244798890038130981149796037, 2.54850094819391397043168801681, 3.58553442918288703961044090947, 4.91509168984274850110358242560, 6.69731407127254663298979864223, 7.13011862715558751042626993019, 7.908142872259586264245154517973, 9.267518806063620923046146306981, 10.41079380855678258898319382037, 11.69806124304747165817865586280, 12.59904716529918380894731506535

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