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Properties

Label 1-1336-1336.83-r0-0-0
Degree $1$
Conductor $1336$
Sign $0.445 + 0.895i$
Analytic cond. $6.20435$
Root an. cond. $6.20435$
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  + (−0.584 + 0.811i)3-s + (0.0567 − 0.998i)5-s + (0.914 + 0.404i)7-s + (−0.316 − 0.948i)9-s + (−0.0189 + 0.999i)11-s + (0.421 + 0.906i)13-s + (0.776 + 0.629i)15-s + (−0.132 − 0.991i)17-s + (0.988 + 0.150i)19-s + (−0.862 + 0.505i)21-s + (0.614 + 0.788i)23-s + (−0.993 − 0.113i)25-s + (0.954 + 0.298i)27-s + (−0.614 + 0.788i)29-s + (0.387 − 0.922i)31-s + ⋯
L(s)  = 1  + (−0.584 + 0.811i)3-s + (0.0567 − 0.998i)5-s + (0.914 + 0.404i)7-s + (−0.316 − 0.948i)9-s + (−0.0189 + 0.999i)11-s + (0.421 + 0.906i)13-s + (0.776 + 0.629i)15-s + (−0.132 − 0.991i)17-s + (0.988 + 0.150i)19-s + (−0.862 + 0.505i)21-s + (0.614 + 0.788i)23-s + (−0.993 − 0.113i)25-s + (0.954 + 0.298i)27-s + (−0.614 + 0.788i)29-s + (0.387 − 0.922i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1336\)    =    \(2^{3} \cdot 167\)
Sign: $0.445 + 0.895i$
Analytic conductor: \(6.20435\)
Root analytic conductor: \(6.20435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1336} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1336,\ (0:\ ),\ 0.445 + 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.191136690 + 0.7381432703i\)
\(L(\frac12)\) \(\approx\) \(1.191136690 + 0.7381432703i\)
\(L(1)\) \(\approx\) \(0.9922492798 + 0.2515571338i\)
\(L(1)\) \(\approx\) \(0.9922492798 + 0.2515571338i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
167 \( 1 \)
good3 \( 1 + (-0.584 + 0.811i)T \)
5 \( 1 + (0.0567 - 0.998i)T \)
7 \( 1 + (0.914 + 0.404i)T \)
11 \( 1 + (-0.0189 + 0.999i)T \)
13 \( 1 + (0.421 + 0.906i)T \)
17 \( 1 + (-0.132 - 0.991i)T \)
19 \( 1 + (0.988 + 0.150i)T \)
23 \( 1 + (0.614 + 0.788i)T \)
29 \( 1 + (-0.614 + 0.788i)T \)
31 \( 1 + (0.387 - 0.922i)T \)
37 \( 1 + (-0.316 + 0.948i)T \)
41 \( 1 + (0.700 + 0.713i)T \)
43 \( 1 + (-0.974 - 0.225i)T \)
47 \( 1 + (-0.351 - 0.936i)T \)
53 \( 1 + (-0.0944 - 0.995i)T \)
59 \( 1 + (-0.132 + 0.991i)T \)
61 \( 1 + (-0.822 - 0.569i)T \)
67 \( 1 + (-0.0567 - 0.998i)T \)
71 \( 1 + (0.553 + 0.832i)T \)
73 \( 1 + (0.942 + 0.334i)T \)
79 \( 1 + (-0.521 + 0.853i)T \)
83 \( 1 + (0.644 + 0.764i)T \)
89 \( 1 + (0.776 - 0.629i)T \)
97 \( 1 + (-0.387 - 0.922i)T \)
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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

−20.859412442914189742474538964400, −19.826289285836123856966808053017, −19.091796307160635290313687998070, −18.44006657756395331568205537746, −17.75123549963543993045386203181, −17.30200926539374427222176481535, −16.32548039682128316407310447613, −15.39719466470521181236901548259, −14.47427255882831011633781763215, −13.85792453021020739863420328866, −13.19555223218589699410911578227, −12.22855645738651113580020727319, −11.27120934674862735176411182997, −10.8670360447526577573594856735, −10.32046225682635639665818572174, −8.79839419371588663153170550853, −7.91459946150924578603522192746, −7.45322453682328423083056920637, −6.39923658026590641117313083458, −5.832512408489045715664581344411, −4.94168595468559267569615843657, −3.655121927803325841636341240710, −2.774239138055702894659702785779, −1.69318652164519343547300677851, −0.708961892787467863802926109913, 1.08454775346311774531959919441, 1.970565088384831901360219637847, 3.41986642573774163360645375371, 4.480689948745482280744418958544, 4.97257736611443360221258451480, 5.54284596613759413752666636523, 6.73271897191803326614799974946, 7.71962632045507020434162852128, 8.752793371497501479571129829435, 9.40720214797477871240745513500, 9.92370460358227175892972763422, 11.32086047069563098134270849203, 11.590574372402162304498308585198, 12.32258383450323300258546501448, 13.37513184216326529352938257866, 14.232164874970047013925106960648, 15.194580343438398970736367697650, 15.682717868025181454296887853978, 16.62065145398939437745939702390, 17.04270943466165699846461410746, 18.02150608501519075246339281788, 18.4105468031417317541035562068, 19.889702411182256362103677760035, 20.49610006679696647414019277773, 21.036528368365896268500667523138

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