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Properties

Label 1-1601-1601.607-r0-0-0
Degree $1$
Conductor $1601$
Sign $0.610 + 0.791i$
Analytic cond. $7.43501$
Root an. cond. $7.43501$
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)2-s + (0.760 + 0.649i)3-s + (−0.309 − 0.951i)4-s + (−0.951 + 0.309i)5-s + (0.972 − 0.233i)6-s + (0.972 + 0.233i)7-s + (−0.951 − 0.309i)8-s + (0.156 + 0.987i)9-s + (−0.309 + 0.951i)10-s + (0.233 + 0.972i)11-s + (0.382 − 0.923i)12-s + (−0.382 + 0.923i)13-s + (0.760 − 0.649i)14-s + (−0.923 − 0.382i)15-s + (−0.809 + 0.587i)16-s + (0.522 − 0.852i)17-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (0.760 + 0.649i)3-s + (−0.309 − 0.951i)4-s + (−0.951 + 0.309i)5-s + (0.972 − 0.233i)6-s + (0.972 + 0.233i)7-s + (−0.951 − 0.309i)8-s + (0.156 + 0.987i)9-s + (−0.309 + 0.951i)10-s + (0.233 + 0.972i)11-s + (0.382 − 0.923i)12-s + (−0.382 + 0.923i)13-s + (0.760 − 0.649i)14-s + (−0.923 − 0.382i)15-s + (−0.809 + 0.587i)16-s + (0.522 − 0.852i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.610 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.610 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1601\)
Sign: $0.610 + 0.791i$
Analytic conductor: \(7.43501\)
Root analytic conductor: \(7.43501\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1601} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1601,\ (0:\ ),\ 0.610 + 0.791i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.910848338 + 0.9394299339i\)
\(L(\frac12)\) \(\approx\) \(1.910848338 + 0.9394299339i\)
\(L(1)\) \(\approx\) \(1.529386484 + 0.001550099829i\)
\(L(1)\) \(\approx\) \(1.529386484 + 0.001550099829i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1601 \( 1 \)
good2 \( 1 + (0.587 - 0.809i)T \)
3 \( 1 + (0.760 + 0.649i)T \)
5 \( 1 + (-0.951 + 0.309i)T \)
7 \( 1 + (0.972 + 0.233i)T \)
11 \( 1 + (0.233 + 0.972i)T \)
13 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 + (0.522 - 0.852i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.760 - 0.649i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (-0.891 + 0.453i)T \)
41 \( 1 + (-0.891 + 0.453i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (-0.891 + 0.453i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.233 + 0.972i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.707 + 0.707i)T \)
71 \( 1 + (-0.649 + 0.760i)T \)
73 \( 1 + (-0.760 + 0.649i)T \)
79 \( 1 + T \)
83 \( 1 + (0.923 + 0.382i)T \)
89 \( 1 + (0.891 - 0.453i)T \)
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

−20.657348733497856378818078579370, −19.42479540279217617749595047623, −19.06185706772702593015130322205, −18.0514424458268915223711742135, −17.15886089511202295932437720384, −16.76594905049472628496555304513, −15.49291397232916673259856006715, −15.063330174296723130796848082013, −14.51370472205364649900966481271, −13.64353453189729835259707564139, −12.99801905495233941645276530819, −12.22557679753224683749902951789, −11.619092745552850089864437741156, −10.62479167862048541249612324624, −9.15726120818785950405272462702, −8.42062199346496883544303278596, −7.84350720797336097207688988922, −7.57894325232700792371757202645, −6.38603276757279556401657335073, −5.58625041592074745358061255722, −4.565203232271030370723899524155, −3.69899944359742230852762250780, −3.19451263600886306411751915503, −1.85659802632243468546368006142, −0.57775769726629147117436335551, 1.42149534621101144929366160790, 2.336769137434557795694714521440, 3.07398371504796421309250587997, 4.02880043061492810562004012665, 4.785379289981565957796092538628, 4.995856352836403479460509256418, 6.78278674620446352524473340133, 7.39636624442485220155565966057, 8.6688040299826411944288861612, 8.99652591179247162605613630717, 10.14107652002090293639935699873, 10.71385938885078632069699619899, 11.59337507683631663092895572267, 12.055582334422632116522840129379, 13.00681404118226331056075568173, 14.09235193843144200665198037045, 14.6121782782306130135327085195, 14.99230807901013204635923826382, 15.750374129364854591557953448054, 16.69764716805089398613098651167, 17.86399066878542589212244936516, 18.715894039983247721482353419737, 19.310194833632125043824672222780, 19.93387288703657432329697549423, 20.73267514875714255566483635149

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