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Properties

Label 1-3968-3968.1341-r0-0-0
Degree $1$
Conductor $3968$
Sign $-0.393 - 0.919i$
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  + (0.418 + 0.908i)3-s + (−0.831 − 0.555i)5-s + (0.649 + 0.760i)7-s + (−0.649 + 0.760i)9-s + (−0.488 − 0.872i)11-s + (−0.346 + 0.938i)13-s + (0.156 − 0.987i)15-s + (−0.891 + 0.453i)17-s + (−0.0392 + 0.999i)19-s + (−0.418 + 0.908i)21-s + (0.760 + 0.649i)23-s + (0.382 + 0.923i)25-s + (−0.962 − 0.271i)27-s + (−0.678 − 0.734i)29-s + (0.587 − 0.809i)33-s + ⋯
L(s)  = 1  + (0.418 + 0.908i)3-s + (−0.831 − 0.555i)5-s + (0.649 + 0.760i)7-s + (−0.649 + 0.760i)9-s + (−0.488 − 0.872i)11-s + (−0.346 + 0.938i)13-s + (0.156 − 0.987i)15-s + (−0.891 + 0.453i)17-s + (−0.0392 + 0.999i)19-s + (−0.418 + 0.908i)21-s + (0.760 + 0.649i)23-s + (0.382 + 0.923i)25-s + (−0.962 − 0.271i)27-s + (−0.678 − 0.734i)29-s + (0.587 − 0.809i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3968\)    =    \(2^{7} \cdot 31\)
Sign: $-0.393 - 0.919i$
Analytic conductor: \(18.4273\)
Root analytic conductor: \(18.4273\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3968} (1341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3968,\ (0:\ ),\ -0.393 - 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08173225836 + 0.1238368615i\)
\(L(\frac12)\) \(\approx\) \(-0.08173225836 + 0.1238368615i\)
\(L(1)\) \(\approx\) \(0.7850590053 + 0.3290167416i\)
\(L(1)\) \(\approx\) \(0.7850590053 + 0.3290167416i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (0.418 + 0.908i)T \)
5 \( 1 + (-0.831 - 0.555i)T \)
7 \( 1 + (0.649 + 0.760i)T \)
11 \( 1 + (-0.488 - 0.872i)T \)
13 \( 1 + (-0.346 + 0.938i)T \)
17 \( 1 + (-0.891 + 0.453i)T \)
19 \( 1 + (-0.0392 + 0.999i)T \)
23 \( 1 + (0.760 + 0.649i)T \)
29 \( 1 + (-0.678 - 0.734i)T \)
37 \( 1 + (0.555 - 0.831i)T \)
41 \( 1 + (-0.852 + 0.522i)T \)
43 \( 1 + (0.418 - 0.908i)T \)
47 \( 1 + (-0.987 - 0.156i)T \)
53 \( 1 + (0.117 - 0.993i)T \)
59 \( 1 + (-0.999 + 0.0392i)T \)
61 \( 1 + (-0.195 + 0.980i)T \)
67 \( 1 + (-0.195 + 0.980i)T \)
71 \( 1 + (0.0784 + 0.996i)T \)
73 \( 1 + (-0.0784 + 0.996i)T \)
79 \( 1 + (-0.453 - 0.891i)T \)
83 \( 1 + (0.0392 - 0.999i)T \)
89 \( 1 + (0.760 - 0.649i)T \)
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

−18.23479473191056933005089007863, −17.501067788883437058151786511101, −16.91842332378985147836227259830, −15.76998635902715063565337088084, −14.98215677923269063943201712880, −14.88209764724687260002997691795, −13.86900537034100157325501013128, −13.2360499175984881663510057792, −12.61995971402271791766483381536, −11.92091694859848427513012571804, −10.97528632602315539930810724282, −10.803141237272198948013020940158, −9.65630568415015777644182754251, −8.7963595487404257206760193567, −7.94919563621912933264637604392, −7.56220753767445691437564474915, −6.96456447627510157224120497329, −6.39707727969970341804868686219, −4.95196458034864948817925793149, −4.62395070808340088214758724298, −3.43671745021530292471841384366, −2.81045387326642122552219224323, −2.07308250776296455839015147818, −0.98530171184446369166027816994, −0.0422645126971259497543280638, 1.563527785575634392369669471133, 2.35402804336346372629674160, 3.31694295288723279608053753906, 4.0264629294513409661784157051, 4.65754887128756054425236323323, 5.37247988658579677097429443095, 6.01870354187000172571939562575, 7.328998623001680154708849730060, 8.01512184014593492128296989965, 8.683276624704805585644600887890, 8.990395837073403170132828371737, 9.88666600184668260185208313773, 10.82502138544258293082112969758, 11.48077633259168367362947033640, 11.76566910074845833571462690617, 12.9062160770557610906111784242, 13.49068263313716684761774494648, 14.52433098306836623145139603867, 14.85654654549512851006495112122, 15.664024255419914148511855797185, 16.06866402414329739318312710204, 16.80716541083363683492844220529, 17.35995944644879007643889003891, 18.551189913241768755697643411539, 19.032129113645714145500462037987

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