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Properties

Label 1-1339-1339.1226-r0-0-0
Degree $1$
Conductor $1339$
Sign $-0.718 - 0.695i$
Analytic cond. $6.21828$
Root an. cond. $6.21828$
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  + (−0.881 − 0.473i)2-s + (−0.696 + 0.717i)3-s + (0.552 + 0.833i)4-s + (−0.932 + 0.361i)5-s + (0.952 − 0.303i)6-s + (−0.816 − 0.577i)7-s + (−0.0922 − 0.995i)8-s + (−0.0307 − 0.999i)9-s + (0.992 + 0.122i)10-s + (−0.881 − 0.473i)11-s + (−0.982 − 0.183i)12-s + (0.445 + 0.895i)14-s + (0.389 − 0.920i)15-s + (−0.389 + 0.920i)16-s + (0.213 + 0.976i)17-s + (−0.445 + 0.895i)18-s + ⋯
L(s)  = 1  + (−0.881 − 0.473i)2-s + (−0.696 + 0.717i)3-s + (0.552 + 0.833i)4-s + (−0.932 + 0.361i)5-s + (0.952 − 0.303i)6-s + (−0.816 − 0.577i)7-s + (−0.0922 − 0.995i)8-s + (−0.0307 − 0.999i)9-s + (0.992 + 0.122i)10-s + (−0.881 − 0.473i)11-s + (−0.982 − 0.183i)12-s + (0.445 + 0.895i)14-s + (0.389 − 0.920i)15-s + (−0.389 + 0.920i)16-s + (0.213 + 0.976i)17-s + (−0.445 + 0.895i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1339\)    =    \(13 \cdot 103\)
Sign: $-0.718 - 0.695i$
Analytic conductor: \(6.21828\)
Root analytic conductor: \(6.21828\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1339} (1226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1339,\ (0:\ ),\ -0.718 - 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01120863957 + 0.02770172255i\)
\(L(\frac12)\) \(\approx\) \(0.01120863957 + 0.02770172255i\)
\(L(1)\) \(\approx\) \(0.3696368476 + 0.03503316186i\)
\(L(1)\) \(\approx\) \(0.3696368476 + 0.03503316186i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.881 - 0.473i)T \)
3 \( 1 + (-0.696 + 0.717i)T \)
5 \( 1 + (-0.932 + 0.361i)T \)
7 \( 1 + (-0.816 - 0.577i)T \)
11 \( 1 + (-0.881 - 0.473i)T \)
17 \( 1 + (0.213 + 0.976i)T \)
19 \( 1 + (0.696 + 0.717i)T \)
23 \( 1 + (-0.0307 + 0.999i)T \)
29 \( 1 + (-0.779 - 0.626i)T \)
31 \( 1 + (0.602 + 0.798i)T \)
37 \( 1 + (-0.332 - 0.943i)T \)
41 \( 1 + (0.153 + 0.988i)T \)
43 \( 1 + (0.332 - 0.943i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.273 + 0.961i)T \)
59 \( 1 + (0.908 + 0.417i)T \)
61 \( 1 + (0.213 + 0.976i)T \)
67 \( 1 + (-0.816 + 0.577i)T \)
71 \( 1 + (0.779 - 0.626i)T \)
73 \( 1 + (-0.932 - 0.361i)T \)
79 \( 1 + (0.932 - 0.361i)T \)
83 \( 1 + (-0.0922 + 0.995i)T \)
89 \( 1 + (-0.552 + 0.833i)T \)
97 \( 1 + (0.952 - 0.303i)T \)
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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

−20.868494696318042007254769326392, −20.17256645303337963313402340220, −19.41383212015551797878532437740, −18.663723312089377172496406293621, −18.374244399271788508716386673398, −17.45059905766718949652397211921, −16.54898890313675235741040044681, −15.98572948071640022026056496276, −15.56529491626069321653584896433, −14.5241534252090685114504473761, −13.302200359984872062388369908642, −12.63099739589437352106343326, −11.76969495335832167343675649431, −11.26080619125845558484648420464, −10.23801904366973548420777778489, −9.42827829363825480868034518863, −8.48492713299753826977698249735, −7.72557915704030712419888502980, −7.085285505777803518786728078486, −6.38883629198895761240454878309, −5.29278741526871345575383066671, −4.833395421290755556566914705210, −3.047107955231515505482834232261, −2.18900587894379451882042710835, −0.81851840502238212816947176397, 0.02650866001017450003478969154, 1.191856796627083753534682942532, 2.885620413867598327028969333, 3.58294996599510161541570509505, 4.10669418530451457207512127886, 5.54408944900374933495295373393, 6.43541364360290509506076585328, 7.382287435544740378105641045072, 8.012254040848882998542380646640, 9.05710117630134795504301587240, 9.99022463275871140999393498850, 10.474514455091739025106645663548, 11.106059553168273464294641274773, 11.900901602410185269126001601002, 12.567980170601696809151284745333, 13.49979336830212834801204954157, 14.86419468043553589207433900004, 15.697077842679843510757659102740, 16.15216121493259106004436852070, 16.72950253374626174516591059518, 17.61356568012648834314062154836, 18.367391515282285262557536436464, 19.22352558432680949577938608136, 19.66735415221206844323834849238, 20.64264554381183838407730148428

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