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Properties

Label 1-1309-1309.774-r0-0-0
Degree $1$
Conductor $1309$
Sign $0.550 - 0.834i$
Analytic cond. $6.07897$
Root an. cond. $6.07897$
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)2-s + (−0.777 + 0.629i)3-s + (−0.913 − 0.406i)4-s + (0.0523 − 0.998i)5-s + (−0.453 − 0.891i)6-s + (0.587 − 0.809i)8-s + (0.207 − 0.978i)9-s + (0.965 + 0.258i)10-s + (0.965 − 0.258i)12-s + (−0.309 − 0.951i)13-s + (0.587 + 0.809i)15-s + (0.669 + 0.743i)16-s + (0.913 + 0.406i)18-s + (−0.406 − 0.913i)19-s + (−0.453 + 0.891i)20-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)2-s + (−0.777 + 0.629i)3-s + (−0.913 − 0.406i)4-s + (0.0523 − 0.998i)5-s + (−0.453 − 0.891i)6-s + (0.587 − 0.809i)8-s + (0.207 − 0.978i)9-s + (0.965 + 0.258i)10-s + (0.965 − 0.258i)12-s + (−0.309 − 0.951i)13-s + (0.587 + 0.809i)15-s + (0.669 + 0.743i)16-s + (0.913 + 0.406i)18-s + (−0.406 − 0.913i)19-s + (−0.453 + 0.891i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1309\)    =    \(7 \cdot 11 \cdot 17\)
Sign: $0.550 - 0.834i$
Analytic conductor: \(6.07897\)
Root analytic conductor: \(6.07897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1309} (774, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1309,\ (0:\ ),\ 0.550 - 0.834i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5205935392 - 0.2803283033i\)
\(L(\frac12)\) \(\approx\) \(0.5205935392 - 0.2803283033i\)
\(L(1)\) \(\approx\) \(0.6236798484 + 0.1491705961i\)
\(L(1)\) \(\approx\) \(0.6236798484 + 0.1491705961i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.207 + 0.978i)T \)
3 \( 1 + (-0.777 + 0.629i)T \)
5 \( 1 + (0.0523 - 0.998i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.406 - 0.913i)T \)
23 \( 1 + (0.258 + 0.965i)T \)
29 \( 1 + (0.987 + 0.156i)T \)
31 \( 1 + (0.998 - 0.0523i)T \)
37 \( 1 + (0.629 - 0.777i)T \)
41 \( 1 + (0.987 - 0.156i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.913 + 0.406i)T \)
53 \( 1 + (-0.743 - 0.669i)T \)
59 \( 1 + (-0.406 + 0.913i)T \)
61 \( 1 + (-0.998 - 0.0523i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.891 - 0.453i)T \)
73 \( 1 + (-0.358 - 0.933i)T \)
79 \( 1 + (-0.838 + 0.544i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.453 - 0.891i)T \)
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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

−21.36310856521590998299639254444, −20.205929788802382030722107969078, −19.23042788366503687948310981, −18.83730369744096675571751393869, −18.31906146863778294731149569782, −17.415849163006136168504684153475, −16.91548670288322274378947606494, −15.93927377921707644214325545497, −14.58603389804428420757254472835, −14.04698684084606197848463594870, −13.233552356517265915826455766743, −12.292339165994804147267208316487, −11.80065147601473789312362272980, −11.02115405792216033277671938207, −10.36793278546745503840582436041, −9.721961172481200172605156155666, −8.47038743751776706619952151144, −7.70687673566470992979370428826, −6.72088484840591297592744676501, −6.09247754311581416931770856301, −4.84463930721576617616181103511, −4.10093562659523326572180565118, −2.83588166105828827504466843275, −2.163855316568560391706886196591, −1.16497304127521897871257962265, 0.33818525365336171106322448561, 1.269366497197993005690726072709, 3.14991778538142786818953749851, 4.48420020716117048994729667252, 4.75592781397332415876713756979, 5.70068393652419612412480777998, 6.27909845848667891551161626278, 7.40893648028026308519438576638, 8.23582681817293238406060570570, 9.14902650630914133273838034645, 9.684910811549775474590388819781, 10.53311771532871555840884206049, 11.48394388167053524662421050923, 12.57016642889330561355768261201, 13.04546578228741988884483550144, 14.06579518578023056813196761681, 15.13114318883292327797137041613, 15.62017660914125659300928228875, 16.26310641507445151163033371982, 17.00887029132285210086762846417, 17.65862020808696470278243787354, 17.97760781181760130011028039297, 19.43981562471597756058148547017, 19.901237553276037426804818405065, 21.21243533317262906694246184347

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