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Properties

Label 1-1617-1617.368-r1-0-0
Degree $1$
Conductor $1617$
Sign $0.998 + 0.0461i$
Analytic cond. $173.770$
Root an. cond. $173.770$
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  + (0.337 − 0.941i)2-s + (−0.772 − 0.635i)4-s + (0.599 + 0.800i)5-s + (−0.858 + 0.512i)8-s + (0.955 − 0.294i)10-s + (0.983 − 0.178i)13-s + (0.193 + 0.981i)16-s + (−0.712 − 0.701i)17-s + (−0.978 − 0.207i)19-s + (0.0448 − 0.998i)20-s + (0.988 − 0.149i)23-s + (−0.280 + 0.959i)25-s + (0.163 − 0.986i)26-s + (0.550 + 0.834i)29-s + (0.913 + 0.406i)31-s + (0.988 + 0.149i)32-s + ⋯
L(s)  = 1  + (0.337 − 0.941i)2-s + (−0.772 − 0.635i)4-s + (0.599 + 0.800i)5-s + (−0.858 + 0.512i)8-s + (0.955 − 0.294i)10-s + (0.983 − 0.178i)13-s + (0.193 + 0.981i)16-s + (−0.712 − 0.701i)17-s + (−0.978 − 0.207i)19-s + (0.0448 − 0.998i)20-s + (0.988 − 0.149i)23-s + (−0.280 + 0.959i)25-s + (0.163 − 0.986i)26-s + (0.550 + 0.834i)29-s + (0.913 + 0.406i)31-s + (0.988 + 0.149i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.998 + 0.0461i$
Analytic conductor: \(173.770\)
Root analytic conductor: \(173.770\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1617,\ (1:\ ),\ 0.998 + 0.0461i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.342399069 + 0.05404557941i\)
\(L(\frac12)\) \(\approx\) \(2.342399069 + 0.05404557941i\)
\(L(1)\) \(\approx\) \(1.213882697 - 0.3902609951i\)
\(L(1)\) \(\approx\) \(1.213882697 - 0.3902609951i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.337 - 0.941i)T \)
5 \( 1 + (0.599 + 0.800i)T \)
13 \( 1 + (0.983 - 0.178i)T \)
17 \( 1 + (-0.712 - 0.701i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (0.988 - 0.149i)T \)
29 \( 1 + (0.550 + 0.834i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (0.998 - 0.0598i)T \)
41 \( 1 + (-0.858 + 0.512i)T \)
43 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (-0.791 - 0.611i)T \)
53 \( 1 + (-0.251 - 0.967i)T \)
59 \( 1 + (-0.0149 + 0.999i)T \)
61 \( 1 + (0.251 - 0.967i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.0448 + 0.998i)T \)
73 \( 1 + (0.791 - 0.611i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (-0.983 - 0.178i)T \)
89 \( 1 + (-0.0747 + 0.997i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−20.55078052581498368577847335862, −19.39092508279465143201306945885, −18.62045037498251470811842848043, −17.72322132779489347435640820932, −17.11348702979366545476692305451, −16.66403226637084359819029378390, −15.655233256866540801085450900420, −15.23326254520981331463339465331, −14.19880555577033601026695784010, −13.43525443311971303231520479243, −13.04763656942169006338320180768, −12.25434286598340283604844962358, −11.23375161390519410809770164711, −10.17052463706485806899534202159, −9.29070853270497453364780764185, −8.51754850890797195188528707270, −8.16900136033230068726365385414, −6.83309996069927750384579080598, −6.2145097728465436794202468080, −5.57054464122005297543170012387, −4.51619539469196926681349241917, −4.074403164366077239800996586470, −2.78952461222113373653026932257, −1.582262857799544325791839410381, −0.4523472480182466679122530621, 0.881666847112729660677094774566, 1.852490835326279082465571099165, 2.78179549932082587684694314616, 3.34227161268536079340081516866, 4.48978641991108787472891690531, 5.213929426071275082620984242, 6.34922725850825769134949238300, 6.71090585327330332693695257977, 8.2125274929966016562668061268, 8.970914014202282323759941401390, 9.76505002217687156431343809646, 10.59213288572379865592148007444, 11.0756558732194811022050966224, 11.76778241110391045452654721446, 13.0024506649256598915743419556, 13.2867721798348318509113571774, 14.13693635424303056488138306024, 14.846736405680670611077909881151, 15.51144116485579407564684585728, 16.65270133328011775406890942519, 17.723652672086280637099910985834, 18.10744810302861622881657065852, 18.82104900152679315441745175713, 19.560671604255748298110657723, 20.32406246706653009594619998479

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