Properties

Label 1-3997-3997.692-r1-0-0
Degree $1$
Conductor $3997$
Sign $0.598 + 0.801i$
Analytic cond. $429.537$
Root an. cond. $429.537$
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  + (−0.998 + 0.0550i)2-s + (−0.930 + 0.366i)3-s + (0.993 − 0.110i)4-s + (−0.471 + 0.882i)5-s + (0.909 − 0.416i)6-s + (−0.986 + 0.164i)8-s + (0.731 − 0.681i)9-s + (0.421 − 0.906i)10-s + (0.996 + 0.0880i)11-s + (−0.884 + 0.466i)12-s + (−0.999 − 0.0220i)13-s + (0.115 − 0.993i)15-s + (0.975 − 0.218i)16-s + (0.660 − 0.750i)17-s + (−0.693 + 0.720i)18-s + (0.0605 + 0.998i)19-s + ⋯
L(s)  = 1  + (−0.998 + 0.0550i)2-s + (−0.930 + 0.366i)3-s + (0.993 − 0.110i)4-s + (−0.471 + 0.882i)5-s + (0.909 − 0.416i)6-s + (−0.986 + 0.164i)8-s + (0.731 − 0.681i)9-s + (0.421 − 0.906i)10-s + (0.996 + 0.0880i)11-s + (−0.884 + 0.466i)12-s + (−0.999 − 0.0220i)13-s + (0.115 − 0.993i)15-s + (0.975 − 0.218i)16-s + (0.660 − 0.750i)17-s + (−0.693 + 0.720i)18-s + (0.0605 + 0.998i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $0.598 + 0.801i$
Analytic conductor: \(429.537\)
Root analytic conductor: \(429.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (692, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3997,\ (1:\ ),\ 0.598 + 0.801i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8826562983 + 0.4424611310i\)
\(L(\frac12)\) \(\approx\) \(0.8826562983 + 0.4424611310i\)
\(L(1)\) \(\approx\) \(0.5248969370 + 0.1685755225i\)
\(L(1)\) \(\approx\) \(0.5248969370 + 0.1685755225i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
571 \( 1 \)
good2 \( 1 + (-0.998 + 0.0550i)T \)
3 \( 1 + (-0.930 + 0.366i)T \)
5 \( 1 + (-0.471 + 0.882i)T \)
11 \( 1 + (0.996 + 0.0880i)T \)
13 \( 1 + (-0.999 - 0.0220i)T \)
17 \( 1 + (0.660 - 0.750i)T \)
19 \( 1 + (0.0605 + 0.998i)T \)
23 \( 1 + (-0.461 + 0.887i)T \)
29 \( 1 + (0.904 + 0.426i)T \)
31 \( 1 + (0.677 + 0.735i)T \)
37 \( 1 + (0.329 + 0.944i)T \)
41 \( 1 + (-0.137 - 0.990i)T \)
43 \( 1 + (0.874 - 0.485i)T \)
47 \( 1 + (0.298 - 0.954i)T \)
53 \( 1 + (0.840 - 0.542i)T \)
59 \( 1 + (0.879 - 0.475i)T \)
61 \( 1 + (0.709 + 0.705i)T \)
67 \( 1 + (-0.889 - 0.456i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (0.982 + 0.186i)T \)
79 \( 1 + (0.583 + 0.812i)T \)
83 \( 1 + (-0.815 - 0.578i)T \)
89 \( 1 + (-0.0935 + 0.995i)T \)
97 \( 1 + (0.868 + 0.495i)T \)
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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

−18.01421416183417865069765529666, −17.427644953238352602455959193927, −16.96250357045478351151095463507, −16.469018242065674596503458275308, −15.83526561129167406702135586283, −15.06345680844670892016256845814, −14.23656156662196497044901741643, −13.01964710005910017132762855413, −12.48624875836260265367623501561, −11.85108991121698599017869885216, −11.51094246944946202432620254741, −10.599016479577684722801150355143, −9.87617017986038147928789280797, −9.234721866828112665462744556315, −8.40290803067471825114394307973, −7.7466268251249403156178439761, −7.123254468574113534196579049903, −6.280362494933410010505164749531, −5.74696011601255427250941404482, −4.63218651365606799834869988886, −4.12864540675436579918092616382, −2.771802950250448467510154844423, −1.889238810007117199457732189906, −0.90561453173143249779230160669, −0.594048622704489565394365519, 0.49368324216530382823695622838, 1.29066670085772342985116078435, 2.35237604375432005820924348919, 3.33683360272367105134075351586, 3.97670021207296910980098170629, 5.12749798635122320571722605753, 5.85769678756687941641348727116, 6.72351709034804771194954288471, 7.09045219258419217583657758602, 7.8055583854240434590198008895, 8.75564330557539396283710052592, 9.73800537362488276197998515943, 10.06637381003592551177096964236, 10.648733842957641464274469294129, 11.65164124202383347586260871871, 11.916616303498494608956571494788, 12.32484827806115504762381706678, 13.92074636614500170606177027467, 14.60208738768495950838570688548, 15.170603541478733187322585034810, 15.94518226263021098543192894441, 16.416696386063434724840976727267, 17.19884666404543468778164383928, 17.658393381150789031668898819182, 18.3294459301799923258336503454

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