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Properties

Label 1-1580-1580.1379-r1-0-0
Degree $1$
Conductor $1580$
Sign $-0.999 + 0.00206i$
Analytic cond. $169.794$
Root an. cond. $169.794$
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  + (0.692 + 0.721i)3-s + (0.278 − 0.960i)7-s + (−0.0402 + 0.999i)9-s + (0.200 + 0.979i)11-s + (0.632 − 0.774i)13-s + (0.354 + 0.935i)17-s + (−0.799 − 0.600i)19-s + (0.885 − 0.464i)21-s + (−0.5 + 0.866i)23-s + (−0.748 + 0.663i)27-s + (−0.845 + 0.534i)29-s + (−0.428 − 0.903i)31-s + (−0.568 + 0.822i)33-s + (0.919 − 0.391i)37-s + (0.996 − 0.0804i)39-s + ⋯
L(s)  = 1  + (0.692 + 0.721i)3-s + (0.278 − 0.960i)7-s + (−0.0402 + 0.999i)9-s + (0.200 + 0.979i)11-s + (0.632 − 0.774i)13-s + (0.354 + 0.935i)17-s + (−0.799 − 0.600i)19-s + (0.885 − 0.464i)21-s + (−0.5 + 0.866i)23-s + (−0.748 + 0.663i)27-s + (−0.845 + 0.534i)29-s + (−0.428 − 0.903i)31-s + (−0.568 + 0.822i)33-s + (0.919 − 0.391i)37-s + (0.996 − 0.0804i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1580 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00206i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1580 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1580\)    =    \(2^{2} \cdot 5 \cdot 79\)
Sign: $-0.999 + 0.00206i$
Analytic conductor: \(169.794\)
Root analytic conductor: \(169.794\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1580} (1379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1580,\ (1:\ ),\ -0.999 + 0.00206i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001027952744 + 0.9974722270i\)
\(L(\frac12)\) \(\approx\) \(0.001027952744 + 0.9974722270i\)
\(L(1)\) \(\approx\) \(1.141391766 + 0.3345058648i\)
\(L(1)\) \(\approx\) \(1.141391766 + 0.3345058648i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
79 \( 1 \)
good3 \( 1 + (0.692 + 0.721i)T \)
7 \( 1 + (0.278 - 0.960i)T \)
11 \( 1 + (0.200 + 0.979i)T \)
13 \( 1 + (0.632 - 0.774i)T \)
17 \( 1 + (0.354 + 0.935i)T \)
19 \( 1 + (-0.799 - 0.600i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.845 + 0.534i)T \)
31 \( 1 + (-0.428 - 0.903i)T \)
37 \( 1 + (0.919 - 0.391i)T \)
41 \( 1 + (-0.748 - 0.663i)T \)
43 \( 1 + (-0.200 + 0.979i)T \)
47 \( 1 + (-0.919 - 0.391i)T \)
53 \( 1 + (-0.692 + 0.721i)T \)
59 \( 1 + (-0.987 + 0.160i)T \)
61 \( 1 + (0.120 - 0.992i)T \)
67 \( 1 + (0.568 + 0.822i)T \)
71 \( 1 + (0.970 + 0.239i)T \)
73 \( 1 + (0.632 + 0.774i)T \)
83 \( 1 + (0.987 + 0.160i)T \)
89 \( 1 + (-0.970 + 0.239i)T \)
97 \( 1 + (-0.120 + 0.992i)T \)
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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

−19.84282968370763300494707578353, −19.0061782809006099112069933885, −18.5123416104628797282912588275, −18.1572073520734539247159439328, −16.87426983741712805774978328862, −16.24572587196391796276418099588, −15.30848748699250780825850367878, −14.5072808600650116335583983824, −14.01349827754315973222191348099, −13.214286068650378374306970297574, −12.36350802055771174830968227021, −11.71411621719359645104495812501, −11.02600121706008026053690242504, −9.73025588478181140909058177686, −8.97475503577444304005868902778, −8.40319903607889666135324393959, −7.77093885373727929316430770244, −6.510340509732248425120432827, −6.17448976363046020616819301661, −5.058334072371597614775570136491, −3.86569467412758676123315743988, −3.071686809524273959190619847656, −2.145085157601084868543006344151, −1.39933665288311678884628394208, −0.15070093003685691305462363852, 1.36083671197279001565755807129, 2.186798632055190836807188466295, 3.46597309182402457990968920766, 3.9655073590434241894702168393, 4.76727872280948863864797500200, 5.72310668866034852835983028274, 6.85977740243259313580038958146, 7.8057277003068580595848239735, 8.21660363361392002559408417192, 9.37660427205824686859680490187, 9.91126082027893002756054559920, 10.7912084640083379527836586714, 11.20711390295947890164142914517, 12.6541509697377222320663751443, 13.196284728028218290290803918581, 13.99990574524136379305938654365, 14.91363878038350824752936629983, 15.17260443155769884041480573918, 16.17056351135888527784490791857, 16.99683492159596085442430969739, 17.51675475339149021851379319400, 18.49849038321502065567992239078, 19.50051971147772235063222043333, 20.146319189300569970641733723259, 20.45186478478449630053573590719

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