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Properties

Label 1-3328-3328.619-r0-0-0
Degree $1$
Conductor $3328$
Sign $0.170 + 0.985i$
Analytic cond. $15.4551$
Root an. cond. $15.4551$
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  + (−0.773 − 0.634i)3-s + (−0.956 + 0.290i)5-s + (−0.195 + 0.980i)7-s + (0.195 + 0.980i)9-s + (−0.995 − 0.0980i)11-s + (0.923 + 0.382i)15-s + (0.382 + 0.923i)17-s + (−0.881 + 0.471i)19-s + (0.773 − 0.634i)21-s + (0.555 − 0.831i)23-s + (0.831 − 0.555i)25-s + (0.471 − 0.881i)27-s + (0.995 − 0.0980i)29-s + (−0.707 + 0.707i)31-s + (0.707 + 0.707i)33-s + ⋯
L(s)  = 1  + (−0.773 − 0.634i)3-s + (−0.956 + 0.290i)5-s + (−0.195 + 0.980i)7-s + (0.195 + 0.980i)9-s + (−0.995 − 0.0980i)11-s + (0.923 + 0.382i)15-s + (0.382 + 0.923i)17-s + (−0.881 + 0.471i)19-s + (0.773 − 0.634i)21-s + (0.555 − 0.831i)23-s + (0.831 − 0.555i)25-s + (0.471 − 0.881i)27-s + (0.995 − 0.0980i)29-s + (−0.707 + 0.707i)31-s + (0.707 + 0.707i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.170 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.170 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.170 + 0.985i$
Analytic conductor: \(15.4551\)
Root analytic conductor: \(15.4551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3328,\ (0:\ ),\ 0.170 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4962350286 + 0.4177666895i\)
\(L(\frac12)\) \(\approx\) \(0.4962350286 + 0.4177666895i\)
\(L(1)\) \(\approx\) \(0.6273290180 + 0.05080121539i\)
\(L(1)\) \(\approx\) \(0.6273290180 + 0.05080121539i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.773 - 0.634i)T \)
5 \( 1 + (-0.956 + 0.290i)T \)
7 \( 1 + (-0.195 + 0.980i)T \)
11 \( 1 + (-0.995 - 0.0980i)T \)
17 \( 1 + (0.382 + 0.923i)T \)
19 \( 1 + (-0.881 + 0.471i)T \)
23 \( 1 + (0.555 - 0.831i)T \)
29 \( 1 + (0.995 - 0.0980i)T \)
31 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (0.471 - 0.881i)T \)
41 \( 1 + (0.555 - 0.831i)T \)
43 \( 1 + (-0.773 + 0.634i)T \)
47 \( 1 + (0.382 + 0.923i)T \)
53 \( 1 + (0.995 + 0.0980i)T \)
59 \( 1 + (-0.290 - 0.956i)T \)
61 \( 1 + (0.634 - 0.773i)T \)
67 \( 1 + (0.773 + 0.634i)T \)
71 \( 1 + (0.980 + 0.195i)T \)
73 \( 1 + (-0.195 - 0.980i)T \)
79 \( 1 + (0.923 + 0.382i)T \)
83 \( 1 + (-0.471 - 0.881i)T \)
89 \( 1 + (0.831 - 0.555i)T \)
97 \( 1 + (-0.707 + 0.707i)T \)
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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

−18.56648860481260061456686476762, −18.01028044654312334274398409757, −17.02373927916921786138276866587, −16.65173922007092383371227746439, −16.01578322936301182731708028955, −15.327982723291175759943145521877, −14.886533870741028074910972762156, −13.68194873595021959108638240214, −13.06834313863558082395981717444, −12.32973818964607464379078716145, −11.51008215963868730539085797569, −11.05736923573728418821845125035, −10.31149614968981429644911544640, −9.726094544704963330424640070109, −8.80688927933380390691910499048, −7.93336664181623240560477451671, −7.1960848618414877568002355380, −6.64654143860862149753094923991, −5.4586484533883289530932398927, −4.88933526000202329889470944155, −4.21965484450622190245101299498, −3.52840287937714270328802879058, −2.67783027408533720326656742058, −1.07233866332046949073560747647, −0.35835876494271064434821508691, 0.74350944773089989008651596271, 2.03086016594475048841939570412, 2.67135796696571757370199477458, 3.65786172381801079182549989695, 4.66144649292351876216743904722, 5.35750879617176300533305051865, 6.16634984400070668322260897955, 6.71893275012526405098732643055, 7.68257864462894782872892010924, 8.204764211230736535789031802319, 8.83183556048329321554262349259, 10.15266360828248856990840475169, 10.759370257607976391837966705421, 11.24829250336846490613041477217, 12.28938362295307534278097259389, 12.507998598468351247801384155637, 13.06061094267394202294003340752, 14.297295708639904845276618612700, 14.878082307364131211287649629050, 15.742218232585637616851428097, 16.140377689407999883402976545125, 16.90955328599391232051106724422, 17.803860448609856414455856108946, 18.39157160708262831602110453396, 19.04234296468182253049014594954

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