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Properties

Label 1-3968-3968.1139-r0-0-0
Degree $1$
Conductor $3968$
Sign $0.840 - 0.542i$
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  + (0.346 − 0.938i)3-s + (−0.980 + 0.195i)5-s + (−0.760 + 0.649i)7-s + (−0.760 − 0.649i)9-s + (0.962 + 0.271i)11-s + (0.908 − 0.418i)13-s + (−0.156 + 0.987i)15-s + (−0.891 + 0.453i)17-s + (0.734 − 0.678i)19-s + (0.346 + 0.938i)21-s + (−0.649 + 0.760i)23-s + (0.923 − 0.382i)25-s + (−0.872 + 0.488i)27-s + (0.0392 − 0.999i)29-s + (0.587 − 0.809i)33-s + ⋯
L(s)  = 1  + (0.346 − 0.938i)3-s + (−0.980 + 0.195i)5-s + (−0.760 + 0.649i)7-s + (−0.760 − 0.649i)9-s + (0.962 + 0.271i)11-s + (0.908 − 0.418i)13-s + (−0.156 + 0.987i)15-s + (−0.891 + 0.453i)17-s + (0.734 − 0.678i)19-s + (0.346 + 0.938i)21-s + (−0.649 + 0.760i)23-s + (0.923 − 0.382i)25-s + (−0.872 + 0.488i)27-s + (0.0392 − 0.999i)29-s + (0.587 − 0.809i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3968\)    =    \(2^{7} \cdot 31\)
Sign: $0.840 - 0.542i$
Analytic conductor: \(18.4273\)
Root analytic conductor: \(18.4273\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3968} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3968,\ (0:\ ),\ 0.840 - 0.542i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.239309988 - 0.3650265575i\)
\(L(\frac12)\) \(\approx\) \(1.239309988 - 0.3650265575i\)
\(L(1)\) \(\approx\) \(0.9193060332 - 0.1980066124i\)
\(L(1)\) \(\approx\) \(0.9193060332 - 0.1980066124i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (0.346 - 0.938i)T \)
5 \( 1 + (-0.980 + 0.195i)T \)
7 \( 1 + (-0.760 + 0.649i)T \)
11 \( 1 + (0.962 + 0.271i)T \)
13 \( 1 + (0.908 - 0.418i)T \)
17 \( 1 + (-0.891 + 0.453i)T \)
19 \( 1 + (0.734 - 0.678i)T \)
23 \( 1 + (-0.649 + 0.760i)T \)
29 \( 1 + (0.0392 - 0.999i)T \)
37 \( 1 + (0.195 + 0.980i)T \)
41 \( 1 + (-0.522 - 0.852i)T \)
43 \( 1 + (0.346 + 0.938i)T \)
47 \( 1 + (-0.987 - 0.156i)T \)
53 \( 1 + (-0.619 - 0.785i)T \)
59 \( 1 + (0.678 - 0.734i)T \)
61 \( 1 + (-0.831 + 0.555i)T \)
67 \( 1 + (-0.831 + 0.555i)T \)
71 \( 1 + (-0.996 + 0.0784i)T \)
73 \( 1 + (0.996 + 0.0784i)T \)
79 \( 1 + (0.453 + 0.891i)T \)
83 \( 1 + (0.734 - 0.678i)T \)
89 \( 1 + (0.649 + 0.760i)T \)
97 \( 1 + (0.951 - 0.309i)T \)
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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

−18.7259073487753707420341676397, −17.896296899667081970106460444635, −16.747335438969512018712763662222, −16.38774814466721491886255964798, −16.02282965531431819047225189858, −15.3069010752635786521180280296, −14.42001520069582726907232476831, −13.96455984489739203908293631408, −13.21712374240360119772250020575, −12.25638363372548592116367306561, −11.586551276197610590418795768207, −10.90971168043117963889658744152, −10.36614298038934745276671420260, −9.32895876857494878637973670317, −8.99972597396046067711197939773, −8.23092150283107635174077856303, −7.4169230900627198744422522004, −6.59705936611453066968877776819, −5.89528259459525388482786398124, −4.68695093427781084415638006488, −4.22349421433281295926176479805, −3.49998787533261047844139085367, −3.143237222699822163356409802496, −1.75306569635863296581578559840, −0.59595075804576855338965767812, 0.61341870106213974226099144157, 1.6077721020668200230388202304, 2.52716521951861870564310775741, 3.39280066022261444209185517227, 3.78721261915237110625980761318, 4.919462917160847614416582723303, 6.17221796452193015295832271091, 6.37508378322642636333718300356, 7.223045403555534246607955642, 7.95531502122984364242166653400, 8.6264787042815365744383609283, 9.1863430895495866901383836178, 10.01952053087326692179596131268, 11.24349603988822349319299307528, 11.64489742529773228307477899426, 12.18597547353213309137179606295, 13.075988068780764010078381141, 13.42958568234523712675495204569, 14.35775294571455995590654246936, 15.16109511287517223096912350244, 15.57059825405071958871114316736, 16.24110967885457630306622646395, 17.27258983939743051700207369210, 17.91661237462000271305917930993, 18.47618551004070854277955758605

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