Properties

Label 1-1009-1009.634-r0-0-0
Degree $1$
Conductor $1009$
Sign $-0.245 - 0.969i$
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  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + 9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + 13-s + 14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + 9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + 13-s + 14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1009\)
Sign: $-0.245 - 0.969i$
Analytic conductor: \(4.68577\)
Root analytic conductor: \(4.68577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1009} (634, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1009,\ (0:\ ),\ -0.245 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8398666410 - 1.078843466i\)
\(L(\frac12)\) \(\approx\) \(0.8398666410 - 1.078843466i\)
\(L(1)\) \(\approx\) \(0.8962950056 - 0.5151133994i\)
\(L(1)\) \(\approx\) \(0.8962950056 - 0.5151133994i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1009 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−22.15454624269161366580663435187, −20.701928922289635552390259386215, −20.16595196120006812722142552893, −19.351611892953011563493657178675, −18.7505296501933203309997610869, −18.06849357605460337807611405899, −17.206542091790983591523217568896, −16.0058203628003105664626360095, −15.63914649782441009260633062366, −14.85345239538599292296713605862, −14.1714314890850750793637843657, −13.39888654736713887594234771432, −12.710705030444391646728579511083, −10.952366491454007246453409943074, −10.51780035723893277663090102271, −9.62560742948866689489852615196, −8.83324879598233795093343568970, −7.868991629459910785608201150939, −7.18571983039049350983598113819, −6.829791639410111766863116492645, −5.54337807280782587454623205379, −4.137926361286499205081362731351, −3.68330085857688876695659488398, −2.39697516116349669573865503785, −1.145545225057148710175778035130, 0.72199491132140321723337923872, 1.7993166673430103460815846984, 3.02543030446116232331441628061, 3.35961437972062399956356609965, 4.534829212215217230154148209875, 5.492640229963999633838625601758, 7.036065015304625351490380889866, 8.005635794857326296425002843903, 8.70194313110743535508206133087, 9.09043049758795972142599547027, 9.88743720823488643851382125911, 11.10905890889562539272559949125, 11.76580794668900549432118046498, 12.75279193031547909636197508916, 13.29412413326781679402374624823, 13.90175597478825371855176539201, 15.36657129392353156289528837563, 15.97447169826756324468078606976, 16.48467752077884604276502856029, 17.8621035029502554715509258048, 18.65179084545895075442380239183, 19.07620377036106747168709970548, 19.844751640162877142367991390158, 20.75089814618216075528001489925, 20.908547909330464969983663201225

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