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Properties

Label 1-4013-4013.3913-r0-0-0
Degree $1$
Conductor $4013$
Sign $0.662 - 0.748i$
Analytic cond. $18.6362$
Root an. cond. $18.6362$
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  + (−0.739 − 0.673i)2-s + (0.273 − 0.961i)3-s + (0.0922 + 0.995i)4-s + (0.602 + 0.798i)5-s + (−0.850 + 0.526i)6-s + (−0.273 + 0.961i)7-s + (0.602 − 0.798i)8-s + (−0.850 − 0.526i)9-s + (0.0922 − 0.995i)10-s + (−0.982 − 0.183i)11-s + (0.982 + 0.183i)12-s + (−0.982 − 0.183i)13-s + (0.850 − 0.526i)14-s + (0.932 − 0.361i)15-s + (−0.982 + 0.183i)16-s + (−0.273 − 0.961i)17-s + ⋯
L(s)  = 1  + (−0.739 − 0.673i)2-s + (0.273 − 0.961i)3-s + (0.0922 + 0.995i)4-s + (0.602 + 0.798i)5-s + (−0.850 + 0.526i)6-s + (−0.273 + 0.961i)7-s + (0.602 − 0.798i)8-s + (−0.850 − 0.526i)9-s + (0.0922 − 0.995i)10-s + (−0.982 − 0.183i)11-s + (0.982 + 0.183i)12-s + (−0.982 − 0.183i)13-s + (0.850 − 0.526i)14-s + (0.932 − 0.361i)15-s + (−0.982 + 0.183i)16-s + (−0.273 − 0.961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4013\)
Sign: $0.662 - 0.748i$
Analytic conductor: \(18.6362\)
Root analytic conductor: \(18.6362\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4013} (3913, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4013,\ (0:\ ),\ 0.662 - 0.748i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6545674064 - 0.2946751545i\)
\(L(\frac12)\) \(\approx\) \(0.6545674064 - 0.2946751545i\)
\(L(1)\) \(\approx\) \(0.6155269656 - 0.2349629034i\)
\(L(1)\) \(\approx\) \(0.6155269656 - 0.2349629034i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4013 \( 1 \)
good2 \( 1 + (-0.739 - 0.673i)T \)
3 \( 1 + (0.273 - 0.961i)T \)
5 \( 1 + (0.602 + 0.798i)T \)
7 \( 1 + (-0.273 + 0.961i)T \)
11 \( 1 + (-0.982 - 0.183i)T \)
13 \( 1 + (-0.982 - 0.183i)T \)
17 \( 1 + (-0.273 - 0.961i)T \)
19 \( 1 + (-0.850 - 0.526i)T \)
23 \( 1 + (-0.932 + 0.361i)T \)
29 \( 1 + (-0.445 - 0.895i)T \)
31 \( 1 + (-0.850 + 0.526i)T \)
37 \( 1 + (0.850 + 0.526i)T \)
41 \( 1 + (0.932 - 0.361i)T \)
43 \( 1 + (-0.273 - 0.961i)T \)
47 \( 1 + (-0.602 + 0.798i)T \)
53 \( 1 + (0.739 + 0.673i)T \)
59 \( 1 + (0.932 + 0.361i)T \)
61 \( 1 + (-0.850 - 0.526i)T \)
67 \( 1 + (0.739 + 0.673i)T \)
71 \( 1 + (0.932 - 0.361i)T \)
73 \( 1 + (0.932 - 0.361i)T \)
79 \( 1 + (0.982 - 0.183i)T \)
83 \( 1 + (-0.850 + 0.526i)T \)
89 \( 1 + (0.739 + 0.673i)T \)
97 \( 1 + (0.739 - 0.673i)T \)
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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

−18.339860599112227936774813517112, −17.69287977585616529921588244139, −16.93385739231840248568263568871, −16.56141579377310321211631828418, −16.18427809421517434077186613341, −15.22775721725028066322523058755, −14.58577891034633315674174865502, −14.13859648282094652940591066090, −13.11417419378801815605518478650, −12.716478842098937296560319069117, −11.31454963112579678047363167313, −10.57013473134748254399481349286, −10.096085861547205611423874679739, −9.64273098026834841254261599769, −8.90475179655845444504911576046, −8.086504730348578733640518795958, −7.71443374096743736968975797625, −6.58656809228341001537353802614, −5.84723874743111712268236964182, −5.128234750058774633792567398738, −4.48260727049657753570698172125, −3.80290259118608107966939148365, −2.35831360712011899924929984627, −1.86666499747512193135305110658, −0.45674593744404772471605384898, 0.48167716768959911254544589108, 1.934760352318213821818771411590, 2.43041346387294021689011190545, 2.693597707832005014436756989440, 3.62521799024824035741802442661, 5.0173695748244223784636190019, 5.869648260226783722958497092367, 6.609148724316034212359478670491, 7.38049687732019896341774531949, 7.86402766681112691327106946612, 8.73978761274188823648086019646, 9.43444282731546622522411711126, 9.93583670721947630339316528519, 10.90311080013407254556565603845, 11.46058334259635015796475949087, 12.23399108601518686090524277289, 12.81631260324490847890463220088, 13.42188380578835295840969678289, 14.082885597893202593402855574691, 15.04134789340733303917230744203, 15.60023530557565191012270307322, 16.610910221210745012829277892750, 17.42469666860345353857922340333, 18.020413827667698592078819914931, 18.352277098692568645301419416542

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