Properties

Degree 1
Conductor 2011
Sign $-0.599 - 0.800i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.566 + 0.824i)2-s + (0.975 − 0.220i)3-s + (−0.359 − 0.933i)4-s + (0.475 − 0.879i)5-s + (−0.370 + 0.928i)6-s + (0.478 + 0.878i)7-s + (0.972 + 0.232i)8-s + (0.903 − 0.429i)9-s + (0.456 + 0.889i)10-s + (−0.682 + 0.730i)11-s + (−0.555 − 0.831i)12-s + (−0.963 − 0.268i)13-s + (−0.994 − 0.102i)14-s + (0.270 − 0.962i)15-s + (−0.742 + 0.670i)16-s + (−0.0453 − 0.998i)17-s + ⋯
L(s,χ)  = 1  + (−0.566 + 0.824i)2-s + (0.975 − 0.220i)3-s + (−0.359 − 0.933i)4-s + (0.475 − 0.879i)5-s + (−0.370 + 0.928i)6-s + (0.478 + 0.878i)7-s + (0.972 + 0.232i)8-s + (0.903 − 0.429i)9-s + (0.456 + 0.889i)10-s + (−0.682 + 0.730i)11-s + (−0.555 − 0.831i)12-s + (−0.963 − 0.268i)13-s + (−0.994 − 0.102i)14-s + (0.270 − 0.962i)15-s + (−0.742 + 0.670i)16-s + (−0.0453 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.599 - 0.800i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.599 - 0.800i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(2011\)
\( \varepsilon \)  =  $-0.599 - 0.800i$
motivic weight  =  \(0\)
character  :  $\chi_{2011} (39, \cdot )$
Sato-Tate  :  $\mu(2010)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 2011,\ (1:\ ),\ -0.599 - 0.800i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3867017561 - 0.7732310467i$
$L(\frac12,\chi)$  $\approx$  $0.3867017561 - 0.7732310467i$
$L(\chi,1)$  $\approx$  1.039428944 + 0.09468280726i
$L(1,\chi)$  $\approx$  1.039428944 + 0.09468280726i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−20.30459315554717218938857025499, −19.06227679296257577886891797100, −18.84213795989996092067341417865, −18.1410047729445842100906072189, −17.04914239176188552927696333515, −16.761663511205989602552295716646, −15.56653660548609671598246438204, −14.60902559700787296228274548160, −14.1759198254557920816474858335, −13.34379670298339260598894512150, −12.89205422973429157412774054459, −11.63454249692244779049202935084, −10.84441035189563808033266435024, −10.30829408849433739984770047264, −9.7949812219453147060804931390, −8.91169205736176426639817729895, −8.00861492881906502080525732884, −7.52896129358123893107318160921, −6.7758231601090810230663648919, −5.32351114690108488063146594528, −4.32291554522667611338038041877, −3.498825079065032684799455628506, −2.88412707345590389543797317003, −2.04072210271286055230164172173, −1.28120639470409163892695636317, 0.14364043322912800266978509442, 1.34736244513154422834794657840, 2.072646364633060304936811857792, 2.886745690186753702833678026769, 4.47057640360848791932152766727, 5.17708814505249728290794796332, 5.572304762958408569545681045246, 7.11177982596786534481960831321, 7.39017757340783340305124192635, 8.335856835528647255888944993609, 8.91933546641205130912326851477, 9.66247208526765589658100821983, 9.9215477405475490031128199714, 11.37599518368024426177044671657, 12.33580586061714677207284152419, 13.12174009298539518822564241494, 13.69978743284097917736724382528, 14.60599834065929449775802281618, 15.1414683631745267536519189317, 15.7817842486928379034191392754, 16.479153152552380680533525951888, 17.610009371411613303620659626032, 17.926607177509139847597334749, 18.57895534749123649175336793537, 19.51819282859827321828328620029

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