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Properties

Label 1-1300-1300.987-r0-0-0
Degree $1$
Conductor $1300$
Sign $0.125 + 0.992i$
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  + (−0.587 − 0.809i)3-s − i·7-s + (−0.309 + 0.951i)9-s + (0.309 + 0.951i)11-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (0.809 − 0.587i)21-s + (−0.951 + 0.309i)23-s + (0.951 − 0.309i)27-s + (0.809 − 0.587i)29-s + (−0.809 − 0.587i)31-s + (0.587 − 0.809i)33-s + (−0.951 − 0.309i)37-s + (−0.309 + 0.951i)41-s − i·43-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)3-s − i·7-s + (−0.309 + 0.951i)9-s + (0.309 + 0.951i)11-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (0.809 − 0.587i)21-s + (−0.951 + 0.309i)23-s + (0.951 − 0.309i)27-s + (0.809 − 0.587i)29-s + (−0.809 − 0.587i)31-s + (0.587 − 0.809i)33-s + (−0.951 − 0.309i)37-s + (−0.309 + 0.951i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.125 + 0.992i$
Analytic conductor: \(6.03717\)
Root analytic conductor: \(6.03717\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (987, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1300,\ (0:\ ),\ 0.125 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6704309691 + 0.5910644073i\)
\(L(\frac12)\) \(\approx\) \(0.6704309691 + 0.5910644073i\)
\(L(1)\) \(\approx\) \(0.8304606543 + 0.05491719803i\)
\(L(1)\) \(\approx\) \(0.8304606543 + 0.05491719803i\)

Euler product

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

−20.82453248592034857235575644708, −20.15816255918617310592099780650, −19.43428839002832102055083916205, −18.435546595755663208257992452232, −17.4520173443964557733421741217, −17.06862682793102002212637733641, −16.06083766826622315981131670430, −15.88083008866052588271073902300, −14.45245233872312237618549672568, −14.17072016471564845350355866496, −13.097793008878563310153008865576, −12.09516438181759274790781302689, −11.406263738855320914893258604455, −10.493020566035218462882637212426, −10.19857311042511101530468267903, −9.08014990395752236353251445505, −8.334152648290963772449259343889, −7.196445241512116813620056605117, −6.38350708289071114988065021115, −5.52359501301145399745185616545, −4.70559863905717708028247423742, −3.67850640166234614020718612736, −3.28193379128015774543155792382, −1.51224126355646896199592448791, −0.40934838360352911272565701592, 1.2607343100241573183586757465, 2.09920181491166510219020457, 3.0134262094382854624602740068, 4.38150107668351623710731757410, 5.38115445236669160700606306021, 5.89576285230403754344681405859, 6.90175539626070582909662485366, 7.625553749239822801735816132797, 8.409678772326684504781920131871, 9.52969392890237111056641520473, 10.13497717484155793171851275716, 11.43593574402739500777256331175, 11.95725316533339129859400358878, 12.40358990762636951451387225672, 13.376618415766190975236745165743, 14.187296862950310640687212744513, 14.98927199701884514755652590955, 15.97276697911151936800418192753, 16.56247241137693822971166948142, 17.60233345840065689117451828049, 18.15657256293703151695400431045, 18.63887068140268460715973142004, 19.577343804761735680906445508512, 20.24861619370973195684755163673, 21.26471395963827827067842277920

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