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Properties

Label 1-1968-1968.509-r0-0-0
Degree $1$
Conductor $1968$
Sign $-0.987 + 0.159i$
Analytic cond. $9.13935$
Root an. cond. $9.13935$
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  + (−0.809 + 0.587i)5-s + (−0.453 − 0.891i)7-s + (−0.156 − 0.987i)11-s + (0.453 − 0.891i)13-s + (−0.156 − 0.987i)17-s + (−0.453 − 0.891i)19-s + (−0.309 − 0.951i)23-s + (0.309 − 0.951i)25-s + (−0.987 − 0.156i)29-s + (0.809 + 0.587i)31-s + (0.891 + 0.453i)35-s + (0.587 + 0.809i)37-s + (0.309 + 0.951i)43-s + (−0.453 + 0.891i)47-s + (−0.587 + 0.809i)49-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)5-s + (−0.453 − 0.891i)7-s + (−0.156 − 0.987i)11-s + (0.453 − 0.891i)13-s + (−0.156 − 0.987i)17-s + (−0.453 − 0.891i)19-s + (−0.309 − 0.951i)23-s + (0.309 − 0.951i)25-s + (−0.987 − 0.156i)29-s + (0.809 + 0.587i)31-s + (0.891 + 0.453i)35-s + (0.587 + 0.809i)37-s + (0.309 + 0.951i)43-s + (−0.453 + 0.891i)47-s + (−0.587 + 0.809i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1968\)    =    \(2^{4} \cdot 3 \cdot 41\)
Sign: $-0.987 + 0.159i$
Analytic conductor: \(9.13935\)
Root analytic conductor: \(9.13935\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1968} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1968,\ (0:\ ),\ -0.987 + 0.159i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03289549661 - 0.4093053829i\)
\(L(\frac12)\) \(\approx\) \(-0.03289549661 - 0.4093053829i\)
\(L(1)\) \(\approx\) \(0.7090176842 - 0.1941112883i\)
\(L(1)\) \(\approx\) \(0.7090176842 - 0.1941112883i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
41 \( 1 \)
good5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.453 - 0.891i)T \)
11 \( 1 + (-0.156 - 0.987i)T \)
13 \( 1 + (0.453 - 0.891i)T \)
17 \( 1 + (-0.156 - 0.987i)T \)
19 \( 1 + (-0.453 - 0.891i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (-0.987 - 0.156i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (-0.453 + 0.891i)T \)
53 \( 1 + (-0.987 - 0.156i)T \)
59 \( 1 + (-0.951 + 0.309i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (-0.156 + 0.987i)T \)
71 \( 1 + (0.987 - 0.156i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.453 + 0.891i)T \)
97 \( 1 + (-0.987 - 0.156i)T \)
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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

−20.28193323858693838441642447479, −19.6306013153180000331733253552, −18.89577636414033306635931758002, −18.43738988026049689923146520805, −17.306782450421108420696251689725, −16.72435974985056301666803499678, −15.86897849967701498588728109109, −15.33711889575069956773693283369, −14.768707854969029966535107138, −13.67349897813273951054496676490, −12.69748338103584412782315065543, −12.40798154449400368330301917442, −11.60841407550908177068035988027, −10.85598898409719660241533618896, −9.6938072546286553852593134869, −9.21966637930350909171298772348, −8.30561659165545926257618092182, −7.733533261781915040110317699208, −6.695696779393797221049673587453, −5.9143236410953484142296808406, −5.05642754678093292056445219665, −4.05536643673869057747309277616, −3.61132761618784603131431320705, −2.19442201787362244431795088136, −1.5665185741182590574660768509, 0.16325819522504242973067155881, 1.012649425342121953170191224443, 2.80840335386793357106389687244, 3.09802405051565486399910827277, 4.125199383273455226927616044152, 4.84379418508281626825337048970, 6.14827264510074254713400329185, 6.64116688503300296957689949695, 7.616700972328931969635415601383, 8.11749566760454643693980606059, 9.07574483050361345214143096668, 10.06926343307968510455100427669, 10.89634950527090597569550929466, 11.14890967840734867456811495096, 12.17935041131464411307525180922, 13.13793993171792496986956391186, 13.63864855520784149029886082695, 14.482992666919137826616954055703, 15.287438113207798151203496178358, 16.06449270733447868692932081022, 16.43846010453474348613424377920, 17.49421059161858796334068276830, 18.23022181544425354185757547610, 18.9635126652265457779201224161, 19.56700627166896429702050102070

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