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Properties

Label 1-2011-2011.39-r1-0-0
Degree $1$
Conductor $2011$
Sign $-0.599 - 0.800i$
Analytic cond. $216.111$
Root an. cond. $216.111$
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  + (−0.566 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $-0.599 - 0.800i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ -0.599 - 0.800i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3867017561 - 0.7732310467i\)
\(L(\frac12)\) \(\approx\) \(0.3867017561 - 0.7732310467i\)
\(L(1)\) \(\approx\) \(1.039428944 + 0.09468280726i\)
\(L(1)\) \(\approx\) \(1.039428944 + 0.09468280726i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (-0.566 + 0.824i)T \)
3 \( 1 + (0.975 - 0.220i)T \)
5 \( 1 + (0.475 - 0.879i)T \)
7 \( 1 + (0.478 + 0.878i)T \)
11 \( 1 + (-0.682 + 0.730i)T \)
13 \( 1 + (-0.963 - 0.268i)T \)
17 \( 1 + (-0.0453 - 0.998i)T \)
19 \( 1 + (0.872 - 0.487i)T \)
23 \( 1 + (0.655 + 0.755i)T \)
29 \( 1 + (-0.894 - 0.446i)T \)
31 \( 1 + (-0.422 + 0.906i)T \)
37 \( 1 + (0.329 - 0.944i)T \)
41 \( 1 + (-0.999 - 0.0281i)T \)
43 \( 1 + (-0.599 + 0.800i)T \)
47 \( 1 + (0.439 - 0.898i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.0609 + 0.998i)T \)
61 \( 1 + (0.997 + 0.0718i)T \)
67 \( 1 + (-0.640 + 0.767i)T \)
71 \( 1 + (0.151 - 0.988i)T \)
73 \( 1 + (-0.984 + 0.174i)T \)
79 \( 1 + (-0.973 + 0.229i)T \)
83 \( 1 + (-0.866 + 0.498i)T \)
89 \( 1 + (-0.995 + 0.0967i)T \)
97 \( 1 + (0.0234 - 0.999i)T \)
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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

−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