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Properties

Label 1-3968-3968.2851-r0-0-0
Degree $1$
Conductor $3968$
Sign $0.427 + 0.903i$
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  + (0.980 + 0.195i)3-s + (−0.555 + 0.831i)5-s + (0.923 − 0.382i)7-s + (0.923 + 0.382i)9-s + (0.195 + 0.980i)11-s + (−0.555 − 0.831i)13-s + (−0.707 + 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.831 + 0.555i)19-s + (0.980 − 0.195i)21-s + (0.382 − 0.923i)23-s + (−0.382 − 0.923i)25-s + (0.831 + 0.555i)27-s + (0.195 − 0.980i)29-s + i·33-s + ⋯
L(s)  = 1  + (0.980 + 0.195i)3-s + (−0.555 + 0.831i)5-s + (0.923 − 0.382i)7-s + (0.923 + 0.382i)9-s + (0.195 + 0.980i)11-s + (−0.555 − 0.831i)13-s + (−0.707 + 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.831 + 0.555i)19-s + (0.980 − 0.195i)21-s + (0.382 − 0.923i)23-s + (−0.382 − 0.923i)25-s + (0.831 + 0.555i)27-s + (0.195 − 0.980i)29-s + i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.427 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.427 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3968\)    =    \(2^{7} \cdot 31\)
Sign: $0.427 + 0.903i$
Analytic conductor: \(18.4273\)
Root analytic conductor: \(18.4273\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3968} (2851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3968,\ (0:\ ),\ 0.427 + 0.903i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.305989759 + 1.460251910i\)
\(L(\frac12)\) \(\approx\) \(2.305989759 + 1.460251910i\)
\(L(1)\) \(\approx\) \(1.508785290 + 0.4030528583i\)
\(L(1)\) \(\approx\) \(1.508785290 + 0.4030528583i\)

Euler product

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

−18.577315009765379658000049262633, −17.73603587170089229235942418295, −16.96184992960540953600845103731, −16.2343064830740335247334506174, −15.652564212537219186082917216177, −14.86026533869422722840062196232, −14.26770989497560967686019942206, −13.76975732868038251511335821842, −12.83718860149856007196748501509, −12.32653245219361402654528306285, −11.473045052643214955379663042473, −11.062316261098998588167214279083, −9.7610741901526268135040976271, −9.06096164653504968283952563626, −8.72835914785700137022992790096, −7.936361533632073033352993430499, −7.43834809815121793055520673672, −6.56930358237747099679734688038, −5.38741891783984672785619796569, −4.821718742439106922723401393497, −4.02646201072330692298754495819, −3.29502154368032030798622259680, −2.35866635353832940343540791446, −1.565017685961882852535420999595, −0.7518707929885622718816668041, 1.06413081443078088859291698663, 2.11278483066850641992753652566, 2.64692958033165443156979951914, 3.6010403261186682969700937814, 4.290795330383028001493622065768, 4.769663011993572544839529039971, 6.012550352018746054763623747265, 6.94529025400833181700309078473, 7.61076172444437760614749318570, 8.05185838437507007244890945874, 8.63361054469129430240102600366, 9.92647060620853515577726922729, 10.17218939570061996851931493582, 10.850491479791163842607253520569, 11.77233019887696566139068639637, 12.526975481442260871549139087, 13.14576831650496771325497088532, 14.23907940190740575611945944271, 14.58006730963416090261158736339, 15.091350785582976124438103793895, 15.47360707613356302866839297394, 16.68488225794302613831433593545, 17.23005898296402470352400557609, 18.130651441146288343203080358073, 18.635734854985288688676626781000

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