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Properties

Label 1-1976-1976.997-r0-0-0
Degree $1$
Conductor $1976$
Sign $-0.700 - 0.713i$
Analytic cond. $9.17650$
Root an. cond. $9.17650$
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  + (0.939 − 0.342i)3-s + (−0.766 − 0.642i)5-s + 7-s + (0.766 − 0.642i)9-s − 11-s + (−0.939 − 0.342i)15-s + (0.173 − 0.984i)17-s + (0.939 − 0.342i)21-s + (−0.939 − 0.342i)23-s + (0.173 + 0.984i)25-s + (0.5 − 0.866i)27-s + (−0.173 − 0.984i)29-s + (−0.5 − 0.866i)31-s + (−0.939 + 0.342i)33-s + (−0.766 − 0.642i)35-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)3-s + (−0.766 − 0.642i)5-s + 7-s + (0.766 − 0.642i)9-s − 11-s + (−0.939 − 0.342i)15-s + (0.173 − 0.984i)17-s + (0.939 − 0.342i)21-s + (−0.939 − 0.342i)23-s + (0.173 + 0.984i)25-s + (0.5 − 0.866i)27-s + (−0.173 − 0.984i)29-s + (−0.5 − 0.866i)31-s + (−0.939 + 0.342i)33-s + (−0.766 − 0.642i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1976 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1976 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1976\)    =    \(2^{3} \cdot 13 \cdot 19\)
Sign: $-0.700 - 0.713i$
Analytic conductor: \(9.17650\)
Root analytic conductor: \(9.17650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1976} (997, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1976,\ (0:\ ),\ -0.700 - 0.713i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6211104016 - 1.479795867i\)
\(L(\frac12)\) \(\approx\) \(0.6211104016 - 1.479795867i\)
\(L(1)\) \(\approx\) \(1.136256992 - 0.5026217305i\)
\(L(1)\) \(\approx\) \(1.136256992 - 0.5026217305i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.939 - 0.342i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + T \)
11 \( 1 - T \)
17 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.173 - 0.984i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.939 - 0.342i)T \)
53 \( 1 + (-0.766 + 0.642i)T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−20.21390557909493195986636985878, −19.58979543231736098855399857572, −18.868025703244461788515520043941, −18.18320530648091061526779317766, −17.56821366648644350853004828932, −16.19188777067175038082585385260, −15.884774730572318130579826754486, −14.86298217724055836583990550268, −14.646364027770045507754331547851, −13.87965862069858486192161952682, −12.926415840228818615829440510659, −12.1877783575378684546094012628, −11.07505120490211678798562158113, −10.67756971106587389064190411706, −9.93279858486970982630077596858, −8.803012219766770428381408509791, −8.110045075467209984948359901241, −7.71542218121579360489763372568, −6.920299405127560401205594354251, −5.629239495194398634288940031697, −4.756398995706076693501120662, −3.94528766144372850118915744087, −3.2579442477347897528079043834, −2.33251383764490290760468975405, −1.51541182045485392602545611489, 0.46021731900085176716955169324, 1.60599557605970994296066780785, 2.43851934389615271554793357993, 3.36846031296153794228426271114, 4.37428360281750265311364770491, 4.87315196051689631701321161301, 5.963726376723185540618183849251, 7.23441511806773362815493846478, 7.9252185385050033178189251064, 8.10965309777984077861436037371, 9.08876597547088162103897446289, 9.83348487771707589616513531484, 10.86262764821905798242914787969, 11.76714097706774024888808375201, 12.258134405486288691838565615095, 13.28163771875448950455738666812, 13.689849073059252676428712558484, 14.67393232625259874651001024714, 15.25480751822089934074486083890, 15.899950801082227400413961312393, 16.69528801133203419000429904095, 17.705382360393858418895431405793, 18.48039068575170236206310039331, 18.85272090360409509405359749980, 19.90984217727469092441213438260

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