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Properties

Label 1-1224-1224.77-r1-0-0
Degree $1$
Conductor $1224$
Sign $0.0407 + 0.999i$
Analytic cond. $131.537$
Root an. cond. $131.537$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 + 0.965i)5-s + (−0.258 − 0.965i)7-s + (0.258 + 0.965i)11-s + (−0.5 − 0.866i)13-s − i·19-s + (0.965 + 0.258i)23-s + (−0.866 − 0.5i)25-s + (0.965 − 0.258i)29-s + (0.258 − 0.965i)31-s + 35-s + (−0.707 − 0.707i)37-s + (−0.965 − 0.258i)41-s + (0.866 + 0.5i)43-s + (−0.5 + 0.866i)47-s + (−0.866 + 0.5i)49-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)5-s + (−0.258 − 0.965i)7-s + (0.258 + 0.965i)11-s + (−0.5 − 0.866i)13-s − i·19-s + (0.965 + 0.258i)23-s + (−0.866 − 0.5i)25-s + (0.965 − 0.258i)29-s + (0.258 − 0.965i)31-s + 35-s + (−0.707 − 0.707i)37-s + (−0.965 − 0.258i)41-s + (0.866 + 0.5i)43-s + (−0.5 + 0.866i)47-s + (−0.866 + 0.5i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0407 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0407 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.0407 + 0.999i$
Analytic conductor: \(131.537\)
Root analytic conductor: \(131.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1224,\ (1:\ ),\ 0.0407 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.036056739 + 0.9946886215i\)
\(L(\frac12)\) \(\approx\) \(1.036056739 + 0.9946886215i\)
\(L(1)\) \(\approx\) \(0.9497034699 + 0.1500002929i\)
\(L(1)\) \(\approx\) \(0.9497034699 + 0.1500002929i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 \)
good5 \( 1 + (-0.258 + 0.965i)T \)
7 \( 1 + (-0.258 - 0.965i)T \)
11 \( 1 + (0.258 + 0.965i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 - iT \)
23 \( 1 + (0.965 + 0.258i)T \)
29 \( 1 + (0.965 - 0.258i)T \)
31 \( 1 + (0.258 - 0.965i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (-0.965 - 0.258i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 - iT \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 + (0.258 + 0.965i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 + (0.258 + 0.965i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + T \)
97 \( 1 + (0.965 - 0.258i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.84723664612979752399465627776, −19.85563124808110693423678674364, −19.22926145307531140002612424348, −18.70518590282790170754710228790, −17.53637903139869298326700968715, −16.884383744811981183604688954878, −16.07481284941485001665458395560, −15.57653665780652304026192151369, −14.600919889794039053782659708021, −13.67420516710420166117402921268, −12.96425936934804050775239986003, −11.993113682011569597933361778060, −11.71330735408093632507901601526, −10.58759389466393795542982544885, −9.42140229099450138256705862711, −8.80509103245127033286903242083, −8.40821214340748823037712804847, −7.05842175210012513725710874813, −6.31363804220332508850134382786, −5.19176412266560727209079229212, −4.74273689513718511156717091388, −3.49483279261092631905596735674, −2.596395747342837372228149397704, −1.42253631841820401429364915985, −0.36546503883242954960867037178, 0.83446089103931188246490273781, 2.12821920390137138033483940552, 3.14522662740534466946673947088, 3.90065487988261047386828552756, 4.81496272035303479068945690825, 5.99334833527019191729672520096, 6.91627158573840999300896454158, 7.43669718453536079405244913851, 8.19878702523620965869822693777, 9.62698515376464529590944388864, 10.14897344349794745148932570317, 10.7893373680749768340069691044, 11.74245833598286894997174491268, 12.578167089226435292656449346132, 13.390842234765953309587689636235, 14.34592721121307366385540201628, 14.852811858551626606129063977757, 15.63583250916510866235070053252, 16.554412920552476091765415250454, 17.56715797780612487821575835874, 17.78061975921450370299531510921, 19.13861248588353019013901238474, 19.38317837593151884847489202940, 20.4157590178703410394973835235, 20.932633647490545355819157024002

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