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Properties

Label 1-1755-1755.1303-r1-0-0
Degree $1$
Conductor $1755$
Sign $-0.937 - 0.347i$
Analytic cond. $188.600$
Root an. cond. $188.600$
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  + (0.984 − 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.642 − 0.766i)7-s + (0.866 − 0.5i)8-s + (0.766 − 0.642i)11-s + (−0.766 − 0.642i)14-s + (0.766 − 0.642i)16-s − i·17-s + (0.5 − 0.866i)19-s + (0.642 − 0.766i)22-s + (−0.642 + 0.766i)23-s + (−0.866 − 0.5i)28-s + (−0.173 − 0.984i)29-s + (0.173 − 0.984i)31-s + (0.642 − 0.766i)32-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.642 − 0.766i)7-s + (0.866 − 0.5i)8-s + (0.766 − 0.642i)11-s + (−0.766 − 0.642i)14-s + (0.766 − 0.642i)16-s − i·17-s + (0.5 − 0.866i)19-s + (0.642 − 0.766i)22-s + (−0.642 + 0.766i)23-s + (−0.866 − 0.5i)28-s + (−0.173 − 0.984i)29-s + (0.173 − 0.984i)31-s + (0.642 − 0.766i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $-0.937 - 0.347i$
Analytic conductor: \(188.600\)
Root analytic conductor: \(188.600\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1755} (1303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1755,\ (1:\ ),\ -0.937 - 0.347i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5792663981 - 3.232931850i\)
\(L(\frac12)\) \(\approx\) \(0.5792663981 - 3.232931850i\)
\(L(1)\) \(\approx\) \(1.639165968 - 0.8004282793i\)
\(L(1)\) \(\approx\) \(1.639165968 - 0.8004282793i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.984 - 0.173i)T \)
7 \( 1 + (-0.642 - 0.766i)T \)
11 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.642 + 0.766i)T \)
29 \( 1 + (-0.173 - 0.984i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (-0.984 + 0.173i)T \)
47 \( 1 + (0.984 - 0.173i)T \)
53 \( 1 + iT \)
59 \( 1 + (-0.766 - 0.642i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (0.342 + 0.939i)T \)
71 \( 1 + T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (0.642 + 0.766i)T \)
89 \( 1 - T \)
97 \( 1 + (0.984 - 0.173i)T \)
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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

−20.37519362476916909930054264464, −19.87543739504201312654227570087, −19.07796872506907013883973482408, −18.20434004408290449216783631932, −17.23802213878035136213157919962, −16.50973332088967208550073775229, −15.861375546130269620150680145619, −15.09047652387105986773048784581, −14.49796704835586116320159032146, −13.80323358006762359711530963104, −12.74304164922392793434623773826, −12.30242295801530248691756603751, −11.82380384458667434007607044782, −10.62724669474474984502074705317, −10.01227790768086916544208865390, −8.89149656724104184748961122578, −8.17828313800245160446745184284, −7.029564741090996893715245453169, −6.50111639047381972082760609512, −5.69023858098468482757420296987, −4.955372739773005948374518090811, −3.84850853606965365743815175126, −3.3651553342647156786193398833, −2.198614652357145472389645827088, −1.49965517171352787507343148517, 0.37533027774329302942028819390, 1.27433964819352402916439844500, 2.49321978035373486840593420265, 3.36980717808757813519885495312, 3.96206606781604381778912026504, 4.886436581465412998593850953143, 5.78043523986080391004700396622, 6.6106244707162663787340734234, 7.16482526418773391508922717972, 8.10249153854799266930262375627, 9.45556839681150634928351038865, 9.89064309192522172626011986836, 10.99086331371208085882478353799, 11.57095395960376985318653351308, 12.203223560242832654757777353330, 13.39044123832739227658631555843, 13.6224143374388183223309147655, 14.2449520337021966989275589201, 15.422323964114075976009212478863, 15.78910903471746557490260524195, 16.77867436684028266975443017502, 17.17932446634595358375394337352, 18.514521429294263195095899774739, 19.22232422416354708429668298239, 20.0182809941656826523975172801

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