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Properties

Label 1-1480-1480.237-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.0702 - 0.997i$
Analytic cond. $6.87309$
Root an. cond. $6.87309$
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  + (−0.642 − 0.766i)3-s + (0.342 − 0.939i)7-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.173 + 0.984i)13-s + (0.173 − 0.984i)17-s + (−0.642 − 0.766i)19-s + (−0.939 + 0.342i)21-s + (0.5 + 0.866i)23-s + (0.866 − 0.5i)27-s + (−0.866 − 0.5i)29-s + i·31-s + (0.984 − 0.173i)33-s + (0.642 − 0.766i)39-s + (−0.173 − 0.984i)41-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)3-s + (0.342 − 0.939i)7-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.173 + 0.984i)13-s + (0.173 − 0.984i)17-s + (−0.642 − 0.766i)19-s + (−0.939 + 0.342i)21-s + (0.5 + 0.866i)23-s + (0.866 − 0.5i)27-s + (−0.866 − 0.5i)29-s + i·31-s + (0.984 − 0.173i)33-s + (0.642 − 0.766i)39-s + (−0.173 − 0.984i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0702 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0702 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.0702 - 0.997i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.0702 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7796665256 - 0.7266748244i\)
\(L(\frac12)\) \(\approx\) \(0.7796665256 - 0.7266748244i\)
\(L(1)\) \(\approx\) \(0.8170654701 - 0.2717536823i\)
\(L(1)\) \(\approx\) \(0.8170654701 - 0.2717536823i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (-0.642 - 0.766i)T \)
7 \( 1 + (0.342 - 0.939i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.173 + 0.984i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (-0.642 - 0.766i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + iT \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.342 - 0.939i)T \)
59 \( 1 + (0.342 + 0.939i)T \)
61 \( 1 + (0.984 - 0.173i)T \)
67 \( 1 + (0.342 - 0.939i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.342 - 0.939i)T \)
83 \( 1 + (0.984 + 0.173i)T \)
89 \( 1 + (0.342 + 0.939i)T \)
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

−20.82490388552817795660569248091, −20.495713211381891846161456437946, −19.08600857012231105456494120508, −18.61270629064514977441861542128, −17.778675516132530609649274434941, −17.003860684325262969060339448452, −16.35963347197677065321817363375, −15.50743306979064760902831283942, −14.984496935192367493892076795840, −14.32141240305635383504863540443, −12.868623275083661324427978473527, −12.58904044927010500073560864329, −11.47310779879384156466330195007, −10.851465011735674986644412809393, −10.291285530579027244095845518913, −9.25820773768089782666261712916, −8.4764795052110410743438427725, −7.85726112552936763605106326006, −6.33216172916735621184223258962, −5.79988087910084338697817371067, −5.220565337628569961126704852264, −4.15905206264333882937492846118, −3.29864281038136908596570178185, −2.339602154948460865072995653863, −0.92140223779599162401554456155, 0.55651189403082506702026674580, 1.66912816843950606443670644860, 2.42429375943152008331814990745, 3.84219858629885455060781558708, 4.77986623248907270641361375361, 5.364583607713960408260699600663, 6.637299879910382394366740204473, 7.16512609885646145273206683204, 7.67950257248032838201341913276, 8.835218532355195557146071629614, 9.77578293533899840884983324030, 10.73879369069274255304156881900, 11.28011822685152070837507440943, 12.05678075272067223219720592429, 12.91136245515333084848847596413, 13.62081385349492754982388354604, 14.15098003376749436408439958562, 15.25224365061331644076356154121, 16.14212188710015518065414825672, 16.90021440701761595732979984362, 17.53459729860189764119604624867, 18.07162610438401171303657912808, 19.006147256036671590869749788434, 19.56074410552114543875085776592, 20.53407674389527552288072426089

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