Properties

Label 1-2009-2009.12-r0-0-0
Degree $1$
Conductor $2009$
Sign $-0.736 - 0.676i$
Analytic cond. $9.32975$
Root an. cond. $9.32975$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.635 + 0.772i)2-s + (−0.757 + 0.652i)3-s + (−0.193 − 0.981i)4-s + (−0.941 + 0.337i)5-s + (−0.0224 − 0.999i)6-s + (0.880 + 0.473i)8-s + (0.149 − 0.988i)9-s + (0.337 − 0.941i)10-s + (−0.999 − 0.00747i)11-s + (0.786 + 0.617i)12-s + (−0.910 − 0.413i)13-s + (0.493 − 0.869i)15-s + (−0.925 + 0.379i)16-s + (0.440 − 0.897i)17-s + (0.669 + 0.743i)18-s + (0.0523 + 0.998i)19-s + ⋯
L(s)  = 1  + (−0.635 + 0.772i)2-s + (−0.757 + 0.652i)3-s + (−0.193 − 0.981i)4-s + (−0.941 + 0.337i)5-s + (−0.0224 − 0.999i)6-s + (0.880 + 0.473i)8-s + (0.149 − 0.988i)9-s + (0.337 − 0.941i)10-s + (−0.999 − 0.00747i)11-s + (0.786 + 0.617i)12-s + (−0.910 − 0.413i)13-s + (0.493 − 0.869i)15-s + (−0.925 + 0.379i)16-s + (0.440 − 0.897i)17-s + (0.669 + 0.743i)18-s + (0.0523 + 0.998i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $-0.736 - 0.676i$
Analytic conductor: \(9.32975\)
Root analytic conductor: \(9.32975\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2009,\ (0:\ ),\ -0.736 - 0.676i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05222799638 + 0.1341807712i\)
\(L(\frac12)\) \(\approx\) \(-0.05222799638 + 0.1341807712i\)
\(L(1)\) \(\approx\) \(0.3594383206 + 0.2134035256i\)
\(L(1)\) \(\approx\) \(0.3594383206 + 0.2134035256i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.635 + 0.772i)T \)
3 \( 1 + (-0.757 + 0.652i)T \)
5 \( 1 + (-0.941 + 0.337i)T \)
11 \( 1 + (-0.999 - 0.00747i)T \)
13 \( 1 + (-0.910 - 0.413i)T \)
17 \( 1 + (0.440 - 0.897i)T \)
19 \( 1 + (0.0523 + 0.998i)T \)
23 \( 1 + (-0.0149 + 0.999i)T \)
29 \( 1 + (-0.928 + 0.372i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (0.992 - 0.119i)T \)
43 \( 1 + (0.722 + 0.691i)T \)
47 \( 1 + (0.995 - 0.0970i)T \)
53 \( 1 + (0.557 - 0.830i)T \)
59 \( 1 + (0.280 + 0.959i)T \)
61 \( 1 + (0.486 - 0.873i)T \)
67 \( 1 + (0.358 - 0.933i)T \)
71 \( 1 + (0.372 - 0.928i)T \)
73 \( 1 + (-0.930 + 0.365i)T \)
79 \( 1 + (-0.258 + 0.965i)T \)
83 \( 1 + (-0.623 + 0.781i)T \)
89 \( 1 + (-0.581 + 0.813i)T \)
97 \( 1 + (-0.987 + 0.156i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.25690783846987168275841446212, −18.92504038312546797845522961565, −18.183410875845999976315665828598, −17.35173433507038557234133445077, −16.802156758573710755276561678669, −16.1466182599314144494718143850, −15.36099467266021790579298564198, −14.21509726696226950240329367469, −13.01298381951991471628260822266, −12.77197204219821369325181570070, −12.030471613986058964109457261860, −11.38963248611128969644362131858, −10.701386981462803881065134443262, −10.07483716677015925995123133035, −8.895663649940619187683515397197, −8.24040416370023734164785023296, −7.39277578461760421206597630520, −7.08750485034418907047343642452, −5.71123855259441862900820473996, −4.74248933309066871734153702478, −4.15550522371468666475604529967, −2.867799821088972104564046072345, −2.15798980659658693229115460649, −1.006163580988831524577608892853, −0.10739073748734259763169455904, 0.882299647981605409664886019716, 2.47965927483255785848523265922, 3.617722592879945346350396484713, 4.51781674048158256396256488664, 5.375488451687566276219422926, 5.80414568794425474760172161340, 7.03534782570964077272317053745, 7.569265980376401784961187925252, 8.16226953449483407022192366638, 9.44949721286846403569508389182, 9.84250930726013083631024884212, 10.71599553809551798087219057858, 11.278871302711688621946190718965, 12.08411505648540798180522911169, 12.984082151055531450688564462669, 14.18696545836895779388979239465, 14.973624250719693331176813181351, 15.352232534414254404429528227376, 16.16751290355270040316340374600, 16.56351224236571078555907623427, 17.377057657817202806790518579597, 18.30296238991459796609472504210, 18.55460153145054283866031532765, 19.54311437605259509492878807468, 20.31598335289035398781327326185

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