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Properties

Label 1-3968-3968.1363-r0-0-0
Degree $1$
Conductor $3968$
Sign $0.740 + 0.671i$
Analytic cond. $18.4273$
Root an. cond. $18.4273$
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  + (−0.555 − 0.831i)3-s + (0.195 + 0.980i)5-s + (−0.382 − 0.923i)7-s + (−0.382 + 0.923i)9-s + (−0.831 − 0.555i)11-s + (0.195 − 0.980i)13-s + (0.707 − 0.707i)15-s + (−0.707 − 0.707i)17-s + (−0.980 − 0.195i)19-s + (−0.555 + 0.831i)21-s + (0.923 + 0.382i)23-s + (−0.923 + 0.382i)25-s + (0.980 − 0.195i)27-s + (−0.831 + 0.555i)29-s + i·33-s + ⋯
L(s)  = 1  + (−0.555 − 0.831i)3-s + (0.195 + 0.980i)5-s + (−0.382 − 0.923i)7-s + (−0.382 + 0.923i)9-s + (−0.831 − 0.555i)11-s + (0.195 − 0.980i)13-s + (0.707 − 0.707i)15-s + (−0.707 − 0.707i)17-s + (−0.980 − 0.195i)19-s + (−0.555 + 0.831i)21-s + (0.923 + 0.382i)23-s + (−0.923 + 0.382i)25-s + (0.980 − 0.195i)27-s + (−0.831 + 0.555i)29-s + i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3968\)    =    \(2^{7} \cdot 31\)
Sign: $0.740 + 0.671i$
Analytic conductor: \(18.4273\)
Root analytic conductor: \(18.4273\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3968} (1363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3968,\ (0:\ ),\ 0.740 + 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4608004856 + 0.1777503618i\)
\(L(\frac12)\) \(\approx\) \(0.4608004856 + 0.1777503618i\)
\(L(1)\) \(\approx\) \(0.6644659785 - 0.1593105526i\)
\(L(1)\) \(\approx\) \(0.6644659785 - 0.1593105526i\)

Euler product

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

−18.41638344446314742181556645039, −17.55463675108332100690824017215, −16.83798259123102445838582041006, −16.56904342088141719873262430679, −15.57802253744632818373701010263, −15.30539781849499352059120708154, −14.61354414329145018283201991414, −13.34492321350807747802738494289, −12.920592092114245488772775599660, −12.21057933138015863093625039092, −11.598211387787933587051276301629, −10.79695533981445013945418391546, −10.05453003220479788281295370240, −9.362948138367595556623652097628, −8.80200416720102087141726557123, −8.302341705119910697066899836271, −7.010750649434932140088701354634, −6.14023411965299297757936291480, −5.71544404599776923575785356173, −4.71045707981815314634591567132, −4.483125082428803744766844294226, −3.47508638574737330177387131501, −2.35496392983019071237499828102, −1.69058120764269344356299745348, −0.21022192923692048452260634833, 0.6808612443805740374808337697, 1.76796847855835712039445311769, 2.80491174736313215391289071211, 3.161388521024637320002521173347, 4.36908828661653582958910706247, 5.31445606050445661385626289627, 5.9933392301677089350779939804, 6.70022607709698422911482820616, 7.27281742523513248087239141678, 7.81800691669348564404852974990, 8.69010444028186955432577646642, 9.81659994380653856620303838989, 10.51491172063236599145783870869, 11.12487679668598181671210055895, 11.30724076659241917751100880507, 12.809694609690672171613202482827, 12.99372328005847295052976218887, 13.66070045182360916809321793689, 14.298961887675274495901230840580, 15.24936854798072651948764256895, 15.84578244992349956079634380079, 16.79392378324722344403382659035, 17.23999506733952343232630211735, 18.06060796988442152919128472767, 18.42567996729144860390011728016

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