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Properties

Label 1-3997-3997.531-r1-0-0
Degree $1$
Conductor $3997$
Sign $-0.908 + 0.418i$
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  + (−0.191 + 0.981i)2-s + (−0.775 − 0.631i)3-s + (−0.926 − 0.376i)4-s + (−0.815 + 0.578i)5-s + (0.768 − 0.639i)6-s + (0.546 − 0.837i)8-s + (0.202 + 0.979i)9-s + (−0.411 − 0.911i)10-s + (0.583 + 0.812i)11-s + (0.480 + 0.876i)12-s + (0.234 − 0.972i)13-s + (0.997 + 0.0660i)15-s + (0.716 + 0.697i)16-s + (−0.884 + 0.466i)17-s + (−0.999 + 0.0110i)18-s + (0.256 + 0.966i)19-s + ⋯
L(s)  = 1  + (−0.191 + 0.981i)2-s + (−0.775 − 0.631i)3-s + (−0.926 − 0.376i)4-s + (−0.815 + 0.578i)5-s + (0.768 − 0.639i)6-s + (0.546 − 0.837i)8-s + (0.202 + 0.979i)9-s + (−0.411 − 0.911i)10-s + (0.583 + 0.812i)11-s + (0.480 + 0.876i)12-s + (0.234 − 0.972i)13-s + (0.997 + 0.0660i)15-s + (0.716 + 0.697i)16-s + (−0.884 + 0.466i)17-s + (−0.999 + 0.0110i)18-s + (0.256 + 0.966i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $-0.908 + 0.418i$
Analytic conductor: \(429.537\)
Root analytic conductor: \(429.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3997,\ (1:\ ),\ -0.908 + 0.418i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1668992778 + 0.7607281955i\)
\(L(\frac12)\) \(\approx\) \(0.1668992778 + 0.7607281955i\)
\(L(1)\) \(\approx\) \(0.5401859666 + 0.2703716245i\)
\(L(1)\) \(\approx\) \(0.5401859666 + 0.2703716245i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
571 \( 1 \)
good2 \( 1 + (-0.191 + 0.981i)T \)
3 \( 1 + (-0.775 - 0.631i)T \)
5 \( 1 + (-0.815 + 0.578i)T \)
11 \( 1 + (0.583 + 0.812i)T \)
13 \( 1 + (0.234 - 0.972i)T \)
17 \( 1 + (-0.884 + 0.466i)T \)
19 \( 1 + (0.256 + 0.966i)T \)
23 \( 1 + (0.0495 - 0.998i)T \)
29 \( 1 + (0.0275 - 0.999i)T \)
31 \( 1 + (-0.245 + 0.969i)T \)
37 \( 1 + (0.761 + 0.648i)T \)
41 \( 1 + (0.298 + 0.954i)T \)
43 \( 1 + (-0.739 + 0.673i)T \)
47 \( 1 + (0.962 - 0.272i)T \)
53 \( 1 + (-0.992 - 0.120i)T \)
59 \( 1 + (0.986 + 0.164i)T \)
61 \( 1 + (-0.224 - 0.974i)T \)
67 \( 1 + (0.391 + 0.920i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (0.942 - 0.335i)T \)
79 \( 1 + (-0.693 + 0.720i)T \)
83 \( 1 + (-0.938 - 0.345i)T \)
89 \( 1 + (0.170 + 0.985i)T \)
97 \( 1 + (-0.0715 + 0.997i)T \)
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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

−18.125816820700647463214437500846, −17.20642090796919685083101284879, −16.85013496745418917899965379845, −16.0435337798404679967479591732, −15.58514236647222509826839044547, −14.56606594996293622686170155566, −13.72923186512551339504929629189, −13.06621343191293154359944830770, −12.23746168208816368641888103868, −11.5579115363456101569235474385, −11.28508112486847509635537482234, −10.7718420748587304323586368714, −9.59780563739901406039597441697, −9.01115119066526706491394346079, −8.81700526431758413842890170808, −7.56228654930124475750817311559, −6.815320225733562196513054162429, −5.71696343514221428163872707744, −4.994106965920597259191168405782, −4.26276372423022298879300612205, −3.82461057266163540774425061524, −3.03955649035699068346923054434, −1.79571312449331365990019585600, −0.83967865006816366873749767515, −0.27563252993765189843634464640, 0.661385516175042218448153219818, 1.51144301745698215173203724576, 2.697347917249667643304304233984, 3.91560926760552424033934085197, 4.47387204708294682082228273616, 5.307288156057197932468789322006, 6.26377452172177328596251748536, 6.59098940534852381545386903434, 7.3023515807080295463064561928, 8.10975209061195465555018976616, 8.34826213384529543283613073354, 9.68725867943994088601218394100, 10.32300827796864982723482849127, 10.990552306951430495676855955470, 11.79682685483914407848952128283, 12.6084344318775511074693783119, 13.00427856081946438724711761216, 14.00845598842100569046743024837, 14.69526960981755922909263436737, 15.25653461501078485318407581209, 15.95837779393828390854643285542, 16.55864595466871367679363440020, 17.37125281720398957700600489377, 17.80352510787968686928677021623, 18.4979224353657636724259012150

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