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Properties

Label 1-1009-1009.101-r0-0-0
Degree $1$
Conductor $1009$
Sign $-0.791 - 0.610i$
Analytic cond. $4.68577$
Root an. cond. $4.68577$
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  + (0.124 + 0.992i)2-s + (0.733 + 0.680i)3-s + (−0.969 + 0.246i)4-s + (0.411 + 0.911i)5-s + (−0.583 + 0.811i)6-s + (0.542 − 0.840i)7-s + (−0.365 − 0.930i)8-s + (0.0747 + 0.997i)9-s + (−0.853 + 0.521i)10-s + (−0.911 + 0.411i)11-s + (−0.878 − 0.478i)12-s + (−0.866 + 0.5i)13-s + (0.900 + 0.433i)14-s + (−0.318 + 0.947i)15-s + (0.878 − 0.478i)16-s + (−0.478 − 0.878i)17-s + ⋯
L(s)  = 1  + (0.124 + 0.992i)2-s + (0.733 + 0.680i)3-s + (−0.969 + 0.246i)4-s + (0.411 + 0.911i)5-s + (−0.583 + 0.811i)6-s + (0.542 − 0.840i)7-s + (−0.365 − 0.930i)8-s + (0.0747 + 0.997i)9-s + (−0.853 + 0.521i)10-s + (−0.911 + 0.411i)11-s + (−0.878 − 0.478i)12-s + (−0.866 + 0.5i)13-s + (0.900 + 0.433i)14-s + (−0.318 + 0.947i)15-s + (0.878 − 0.478i)16-s + (−0.478 − 0.878i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1009\)
Sign: $-0.791 - 0.610i$
Analytic conductor: \(4.68577\)
Root analytic conductor: \(4.68577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1009} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1009,\ (0:\ ),\ -0.791 - 0.610i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4592565623 + 1.346937111i\)
\(L(\frac12)\) \(\approx\) \(-0.4592565623 + 1.346937111i\)
\(L(1)\) \(\approx\) \(0.6579525861 + 1.004145273i\)
\(L(1)\) \(\approx\) \(0.6579525861 + 1.004145273i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1009 \( 1 \)
good2 \( 1 + (0.124 + 0.992i)T \)
3 \( 1 + (0.733 + 0.680i)T \)
5 \( 1 + (0.411 + 0.911i)T \)
7 \( 1 + (0.542 - 0.840i)T \)
11 \( 1 + (-0.911 + 0.411i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (-0.478 - 0.878i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.149 + 0.988i)T \)
29 \( 1 + (-0.456 + 0.889i)T \)
31 \( 1 + (-0.992 - 0.124i)T \)
37 \( 1 + (0.939 + 0.342i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (-0.294 - 0.955i)T \)
53 \( 1 + (0.603 + 0.797i)T \)
59 \( 1 + (-0.433 - 0.900i)T \)
61 \( 1 + (-0.680 - 0.733i)T \)
67 \( 1 + (0.542 + 0.840i)T \)
71 \( 1 + (0.124 + 0.992i)T \)
73 \( 1 + (-0.563 + 0.826i)T \)
79 \( 1 + (0.947 + 0.318i)T \)
83 \( 1 + (0.521 - 0.853i)T \)
89 \( 1 + (0.811 + 0.583i)T \)
97 \( 1 + (0.715 + 0.698i)T \)
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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

−21.08371614621401008896364390600, −20.30817642993908111491920276072, −19.807333249695962147079581743893, −18.90949647223242553294743684579, −18.11441366542944200016643540478, −17.73514533263546463856212034890, −16.64389413889616910448656322994, −15.24388713136090212179487980493, −14.76045426227905854590108614927, −13.64507742521945761707775596803, −13.14152334737967011410714243596, −12.46981080611935852504070578883, −11.86092733242524632131262141422, −10.78497495343239361233371871751, −9.726194942853991769386765067341, −9.036475752487302519061316783539, −8.33059207696030290650893326799, −7.73513137583817518833800825375, −6.07465254832670418606744804000, −5.28428835102419505728507337697, −4.50358284808896982488116149000, −3.17337309738644207280616185816, −2.34553342338384439220382849341, −1.74797065676135816017784595100, −0.49226702022503345821614549239, 1.84937398104984311600250929908, 3.02735058084224014542440513099, 3.83048208831743930330563146021, 4.930396806708529792226501350875, 5.36293266112586503875272191408, 6.94913140535766580837439577656, 7.377766157912451291300016146424, 8.04373148218290487409146808633, 9.34355401375698315256639908851, 9.80062256090819925556648111878, 10.60265583185428644625938980354, 11.609567407171027786490028832117, 13.165185808864485226579098856839, 13.6733802557288772785677564009, 14.41572469555113284628260184385, 14.87461575902392429717913298110, 15.68601995687094369625140843215, 16.519143734619400575935413652462, 17.255108993893578778045862771001, 18.2173792057222369074268760197, 18.650248795675073748432092757052, 19.93340810779610483489131438821, 20.54921231125010337941899387231, 21.72608091014272193274275701465, 21.883944535439646439726094828784

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