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Properties

Label 1-1009-1009.10-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: $\chi_{1009} (10, \cdot )$
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

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

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