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Properties

Label 1-1300-1300.867-r0-0-0
Degree $1$
Conductor $1300$
Sign $-0.994 + 0.100i$
Analytic cond. $6.03717$
Root an. cond. $6.03717$
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  + (−0.207 − 0.978i)3-s + (0.866 + 0.5i)7-s + (−0.913 + 0.406i)9-s + (−0.913 − 0.406i)11-s + (0.207 − 0.978i)17-s + (−0.978 − 0.207i)19-s + (0.309 − 0.951i)21-s + (−0.406 + 0.913i)23-s + (0.587 + 0.809i)27-s + (0.978 − 0.207i)29-s + (−0.309 − 0.951i)31-s + (−0.207 + 0.978i)33-s + (−0.994 − 0.104i)37-s + (−0.104 + 0.994i)41-s + (−0.866 − 0.5i)43-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)3-s + (0.866 + 0.5i)7-s + (−0.913 + 0.406i)9-s + (−0.913 − 0.406i)11-s + (0.207 − 0.978i)17-s + (−0.978 − 0.207i)19-s + (0.309 − 0.951i)21-s + (−0.406 + 0.913i)23-s + (0.587 + 0.809i)27-s + (0.978 − 0.207i)29-s + (−0.309 − 0.951i)31-s + (−0.207 + 0.978i)33-s + (−0.994 − 0.104i)37-s + (−0.104 + 0.994i)41-s + (−0.866 − 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.994 + 0.100i$
Analytic conductor: \(6.03717\)
Root analytic conductor: \(6.03717\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (867, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1300,\ (0:\ ),\ -0.994 + 0.100i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02459011995 - 0.4881670338i\)
\(L(\frac12)\) \(\approx\) \(0.02459011995 - 0.4881670338i\)
\(L(1)\) \(\approx\) \(0.7478215731 - 0.2990130222i\)
\(L(1)\) \(\approx\) \(0.7478215731 - 0.2990130222i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.207 - 0.978i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.913 - 0.406i)T \)
17 \( 1 + (0.207 - 0.978i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (-0.406 + 0.913i)T \)
29 \( 1 + (0.978 - 0.207i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.994 - 0.104i)T \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (-0.951 - 0.309i)T \)
59 \( 1 + (0.913 - 0.406i)T \)
61 \( 1 + (-0.104 - 0.994i)T \)
67 \( 1 + (-0.743 - 0.669i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (0.587 + 0.809i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.951 + 0.309i)T \)
89 \( 1 + (-0.913 - 0.406i)T \)
97 \( 1 + (-0.743 + 0.669i)T \)
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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

−21.11573860605772588674902594949, −20.93663237036764627826583126306, −20.025452460635812159112357629924, −19.23024994578425546553143070985, −18.03986085944656604378103201854, −17.560987837274500042078874834796, −16.76494487310853305285778094407, −16.06251818788423174752461019809, −15.19306070340426832860754689824, −14.6186574163154378234073119487, −13.925170291284396830588971664622, −12.7706825016390815882623920755, −12.055986187854497011460643308393, −10.98361924182780811002889398034, −10.44336708340501591551821818080, −10.01750195546287509685093498062, −8.51802416024852964480396829222, −8.37196146320033892992031563741, −7.10959879995661834231027093974, −6.1038370793586618051016383659, −5.12918328704337254484183967594, −4.53177336555853723467996711917, −3.74271362248599482624943198172, −2.625785985513345929408279188363, −1.50518282781607759658208734450, 0.18638180140631111491328638071, 1.56123598598296328435244105885, 2.33136029775851796986417997639, 3.22306815392967527522696852797, 4.75228312455742706456357513305, 5.37098299644800145055969828947, 6.216035428224000662525782620095, 7.15182849340058345984917467347, 8.06130835882216616319423995515, 8.401092947750986035272059811143, 9.563123054915081286044410375266, 10.69203712378984868304987123448, 11.43690129429411949745244201379, 11.971343513475278236718022843145, 12.91778037699427796413694106831, 13.59023963469538982696926036225, 14.2925772344364740548682549648, 15.19611764069382413649601884416, 16.01272243240313130442746785466, 16.97427004974481002793151774493, 17.738062279219040854392507800734, 18.29240691324942570156584883312, 18.91025155897139946375402231625, 19.68281928708302785669661838734, 20.6390576330992793161018355367

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