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Properties

Label 1-51e2-2601.209-r0-0-0
Degree $1$
Conductor $2601$
Sign $0.999 + 0.00241i$
Analytic cond. $12.0789$
Root an. cond. $12.0789$
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  + (0.997 + 0.0769i)2-s + (0.988 + 0.153i)4-s + (0.994 − 0.0999i)5-s + (0.00769 − 0.999i)7-s + (0.973 + 0.228i)8-s + (0.999 − 0.0230i)10-s + (−0.281 + 0.959i)11-s + (0.999 − 0.0307i)13-s + (0.0845 − 0.996i)14-s + (0.952 + 0.303i)16-s + (0.824 − 0.565i)19-s + (0.998 + 0.0538i)20-s + (−0.354 + 0.935i)22-s + (0.493 + 0.869i)23-s + (0.980 − 0.198i)25-s + (0.998 + 0.0461i)26-s + ⋯
L(s)  = 1  + (0.997 + 0.0769i)2-s + (0.988 + 0.153i)4-s + (0.994 − 0.0999i)5-s + (0.00769 − 0.999i)7-s + (0.973 + 0.228i)8-s + (0.999 − 0.0230i)10-s + (−0.281 + 0.959i)11-s + (0.999 − 0.0307i)13-s + (0.0845 − 0.996i)14-s + (0.952 + 0.303i)16-s + (0.824 − 0.565i)19-s + (0.998 + 0.0538i)20-s + (−0.354 + 0.935i)22-s + (0.493 + 0.869i)23-s + (0.980 − 0.198i)25-s + (0.998 + 0.0461i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $0.999 + 0.00241i$
Analytic conductor: \(12.0789\)
Root analytic conductor: \(12.0789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2601,\ (0:\ ),\ 0.999 + 0.00241i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.636833419 + 0.005600557050i\)
\(L(\frac12)\) \(\approx\) \(4.636833419 + 0.005600557050i\)
\(L(1)\) \(\approx\) \(2.523995226 + 0.003634250688i\)
\(L(1)\) \(\approx\) \(2.523995226 + 0.003634250688i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.997 + 0.0769i)T \)
5 \( 1 + (0.994 - 0.0999i)T \)
7 \( 1 + (0.00769 - 0.999i)T \)
11 \( 1 + (-0.281 + 0.959i)T \)
13 \( 1 + (0.999 - 0.0307i)T \)
19 \( 1 + (0.824 - 0.565i)T \)
23 \( 1 + (0.493 + 0.869i)T \)
29 \( 1 + (0.479 + 0.877i)T \)
31 \( 1 + (-0.935 - 0.354i)T \)
37 \( 1 + (-0.955 - 0.295i)T \)
41 \( 1 + (0.829 + 0.558i)T \)
43 \( 1 + (-0.842 + 0.539i)T \)
47 \( 1 + (0.122 - 0.992i)T \)
53 \( 1 + (-0.990 - 0.138i)T \)
59 \( 1 + (-0.513 - 0.858i)T \)
61 \( 1 + (0.802 - 0.596i)T \)
67 \( 1 + (-0.650 + 0.759i)T \)
71 \( 1 + (-0.967 + 0.251i)T \)
73 \( 1 + (-0.940 + 0.339i)T \)
79 \( 1 + (-0.744 + 0.667i)T \)
83 \( 1 + (0.198 + 0.980i)T \)
89 \( 1 + (0.526 - 0.850i)T \)
97 \( 1 + (-0.712 + 0.701i)T \)
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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

−19.262763457871471534068485574669, −18.72320779656404111865748010975, −18.093345050103705974797139873383, −17.14585338989756816268570727157, −16.17369499818230293726529145722, −15.92964351263854240210650728106, −14.92173419258085194439249408577, −14.267077995528628836882309623103, −13.65280554170856270959376037599, −13.081484514032490808968808147389, −12.30613894285362220144359333280, −11.58688162236934237954052908741, −10.78410204121056280523055098143, −10.24184187649303727481849322975, −9.13405360067236841700769818893, −8.54553564438808826164578037491, −7.51221084023108285129840662195, −6.41725085007355040544400888511, −5.94544658511138212389892790953, −5.457426404323319493396130440234, −4.60828405449936267903517957533, −3.37568814823703429503534768731, −2.91141163403140847175362583959, −1.99049510191978712877454381995, −1.19010755851907666515434136675, 1.25141488020158480210453772860, 1.773502723023775809069172889144, 2.93573289359361350816530424819, 3.60797702690757133261681191104, 4.601716275419165188172843683986, 5.19897968237700757274818063395, 5.94407857178288312315612457420, 6.9439239403220049978000202140, 7.20832804525579429126594355919, 8.29847832718320653569277031695, 9.40828533772184901084502115029, 10.09189366182009176970796719783, 10.84951160560814415026022886443, 11.40458629495681031624821567288, 12.5209096039649727008653033236, 13.15462005053367961452290827714, 13.544435118393308634293133713428, 14.27245891195819858202339401797, 14.88859509891982267425235220168, 15.90616463389728877858856766304, 16.30522624503524267691668869595, 17.337291629092485741281734984124, 17.66553069565178116397744432141, 18.58348553407190476200244360845, 19.78368552854814730295819359699

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