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Properties

Label 1-3968-3968.1125-r0-0-0
Degree $1$
Conductor $3968$
Sign $-0.495 + 0.868i$
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  + (0.845 − 0.533i)3-s + (−0.751 − 0.659i)5-s + (−0.430 − 0.902i)7-s + (0.430 − 0.902i)9-s + (−0.928 + 0.370i)11-s + (0.955 − 0.296i)13-s + (−0.987 − 0.156i)15-s + (−0.544 + 0.838i)17-s + (−0.0915 − 0.995i)19-s + (−0.845 − 0.533i)21-s + (−0.0784 − 0.996i)23-s + (0.130 + 0.991i)25-s + (−0.117 − 0.993i)27-s + (0.346 + 0.938i)29-s + (−0.587 + 0.809i)33-s + ⋯
L(s)  = 1  + (0.845 − 0.533i)3-s + (−0.751 − 0.659i)5-s + (−0.430 − 0.902i)7-s + (0.430 − 0.902i)9-s + (−0.928 + 0.370i)11-s + (0.955 − 0.296i)13-s + (−0.987 − 0.156i)15-s + (−0.544 + 0.838i)17-s + (−0.0915 − 0.995i)19-s + (−0.845 − 0.533i)21-s + (−0.0784 − 0.996i)23-s + (0.130 + 0.991i)25-s + (−0.117 − 0.993i)27-s + (0.346 + 0.938i)29-s + (−0.587 + 0.809i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3968\)    =    \(2^{7} \cdot 31\)
Sign: $-0.495 + 0.868i$
Analytic conductor: \(18.4273\)
Root analytic conductor: \(18.4273\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3968} (1125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3968,\ (0:\ ),\ -0.495 + 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2460890995 - 0.4235476855i\)
\(L(\frac12)\) \(\approx\) \(-0.2460890995 - 0.4235476855i\)
\(L(1)\) \(\approx\) \(0.8315320827 - 0.4654784431i\)
\(L(1)\) \(\approx\) \(0.8315320827 - 0.4654784431i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (0.845 - 0.533i)T \)
5 \( 1 + (-0.751 - 0.659i)T \)
7 \( 1 + (-0.430 - 0.902i)T \)
11 \( 1 + (-0.928 + 0.370i)T \)
13 \( 1 + (0.955 - 0.296i)T \)
17 \( 1 + (-0.544 + 0.838i)T \)
19 \( 1 + (-0.0915 - 0.995i)T \)
23 \( 1 + (-0.0784 - 0.996i)T \)
29 \( 1 + (0.346 + 0.938i)T \)
37 \( 1 + (-0.321 - 0.946i)T \)
41 \( 1 + (-0.688 + 0.725i)T \)
43 \( 1 + (-0.884 + 0.465i)T \)
47 \( 1 + (0.156 - 0.987i)T \)
53 \( 1 + (-0.969 - 0.246i)T \)
59 \( 1 + (0.577 - 0.816i)T \)
61 \( 1 + (-0.555 + 0.831i)T \)
67 \( 1 + (-0.442 - 0.896i)T \)
71 \( 1 + (-0.942 + 0.333i)T \)
73 \( 1 + (-0.182 - 0.983i)T \)
79 \( 1 + (0.838 + 0.544i)T \)
83 \( 1 + (0.816 - 0.577i)T \)
89 \( 1 + (-0.0784 + 0.996i)T \)
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

−18.99446945824173111060109229117, −18.54979797255982204258274808472, −17.8626044392467146346089016568, −16.549716334570699050316752153147, −15.962753567163327210812725692264, −15.46776488231307999494187812016, −15.20128428215106595349660484127, −14.11617537351904568951428466043, −13.66477744651750526909221260778, −12.95085066424993512583448123737, −11.9258702417816543095465168668, −11.40836829344461775599112380, −10.57662430526932009125054467605, −9.978079713738343559061477984962, −9.17927706018159296954503244927, −8.42823340261870322444401206163, −8.00262026543757630876864350024, −7.17479447583339278147661944252, −6.281003889909692840838058715918, −5.49698745581378023610580121640, −4.5928556812343141140301207976, −3.726351925522022021330716779450, −3.140088098093096691837435150078, −2.5803964082626530558551913095, −1.645825956406162418375378915083, 0.12593606306873350178269906142, 1.054092496271830600606712598341, 1.92654094380488942325983340522, 3.000980081894862268549160275490, 3.59775151013570006535422227332, 4.35161126205741084548263122925, 5.03993891858977005064518107103, 6.33126803358959845600811984941, 6.890816140717601819650972369140, 7.637358485117639813448659736448, 8.317697696678698325699328102418, 8.72195497853096332512433679793, 9.59354190880481378925815741404, 10.55459794016504079492048372683, 10.983190189232199690424584524684, 12.1425374252268171789420223721, 12.75859496280657647280117024845, 13.26406962824096302042927603739, 13.639424125528637357401059894830, 14.79053750989113478626794674356, 15.24486053344921635279814966737, 16.03632279581014229014348905237, 16.46628533404074539752534191360, 17.550216569935628280866263602533, 18.052174914851418793374913479816

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