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Properties

Label 1-3997-3997.846-r1-0-0
Degree $1$
Conductor $3997$
Sign $-0.713 - 0.700i$
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  + (0.137 + 0.990i)2-s + (−0.952 + 0.303i)3-s + (−0.962 + 0.272i)4-s + (0.126 − 0.991i)5-s + (−0.431 − 0.901i)6-s + (−0.401 − 0.915i)8-s + (0.815 − 0.578i)9-s + (0.999 − 0.0110i)10-s + (0.509 − 0.860i)11-s + (0.834 − 0.551i)12-s + (0.256 + 0.966i)13-s + (0.180 + 0.983i)15-s + (0.851 − 0.523i)16-s + (0.234 + 0.972i)17-s + (0.685 + 0.728i)18-s + (0.949 + 0.314i)19-s + ⋯
L(s)  = 1  + (0.137 + 0.990i)2-s + (−0.952 + 0.303i)3-s + (−0.962 + 0.272i)4-s + (0.126 − 0.991i)5-s + (−0.431 − 0.901i)6-s + (−0.401 − 0.915i)8-s + (0.815 − 0.578i)9-s + (0.999 − 0.0110i)10-s + (0.509 − 0.860i)11-s + (0.834 − 0.551i)12-s + (0.256 + 0.966i)13-s + (0.180 + 0.983i)15-s + (0.851 − 0.523i)16-s + (0.234 + 0.972i)17-s + (0.685 + 0.728i)18-s + (0.949 + 0.314i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $-0.713 - 0.700i$
Analytic conductor: \(429.537\)
Root analytic conductor: \(429.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (846, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3997,\ (1:\ ),\ -0.713 - 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04687721590 + 0.1145889182i\)
\(L(\frac12)\) \(\approx\) \(-0.04687721590 + 0.1145889182i\)
\(L(1)\) \(\approx\) \(0.7139977661 + 0.3089767877i\)
\(L(1)\) \(\approx\) \(0.7139977661 + 0.3089767877i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
571 \( 1 \)
good2 \( 1 + (0.137 + 0.990i)T \)
3 \( 1 + (-0.952 + 0.303i)T \)
5 \( 1 + (0.126 - 0.991i)T \)
11 \( 1 + (0.509 - 0.860i)T \)
13 \( 1 + (0.256 + 0.966i)T \)
17 \( 1 + (0.234 + 0.972i)T \)
19 \( 1 + (0.949 + 0.314i)T \)
23 \( 1 + (0.863 - 0.504i)T \)
29 \( 1 + (0.451 + 0.892i)T \)
31 \( 1 + (0.879 + 0.475i)T \)
37 \( 1 + (-0.360 - 0.932i)T \)
41 \( 1 + (-0.904 - 0.426i)T \)
43 \( 1 + (-0.319 - 0.947i)T \)
47 \( 1 + (-0.0275 - 0.999i)T \)
53 \( 1 + (-0.899 - 0.436i)T \)
59 \( 1 + (-0.945 - 0.324i)T \)
61 \( 1 + (-0.528 - 0.849i)T \)
67 \( 1 + (0.0715 + 0.997i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (-0.988 - 0.153i)T \)
79 \( 1 + (-0.989 + 0.142i)T \)
83 \( 1 + (-0.565 + 0.824i)T \)
89 \( 1 + (0.997 + 0.0770i)T \)
97 \( 1 + (0.556 + 0.831i)T \)
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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

−17.961291054262546365268780752, −17.51789208657770550797626410095, −16.91176257232109292407589167612, −15.60734177059436341145995752877, −15.25421896506599112052912611042, −14.26581248045976456418253839898, −13.55167530149134059187314430038, −13.10506298690531156611566736792, −12.09919085158061432283749874184, −11.68298923476165792210717145067, −11.17597844443958380657196665515, −10.31680228286546512863328242188, −9.901211122479141080620692816846, −9.26018890046875563392968199456, −7.89819991474518630587656184872, −7.399115455617646499029522146865, −6.4510735035905092372089027347, −5.84536356712213218071685565428, −4.91091206210521144810041682536, −4.46544172370197713665430323069, −3.14096678494510820946495849418, −2.872664604574793825497533117, −1.65347296119792209572206488822, −1.03755625558381007168335259228, −0.02577916706356091951924476758, 0.94547111850979609042905676693, 1.522755145720787049699229185143, 3.41748669281281944902945599728, 3.94778805261719689916121874673, 4.82064519740373688372568300574, 5.29398708075091682821042846818, 6.03325138170049860476134911181, 6.61041554780689346810474622122, 7.337557758287405888106139216251, 8.58500904039493400538652149604, 8.71395733384047467080971530556, 9.62185966270866777180002491242, 10.32558182398463816752408034609, 11.279367014855210110191608012208, 12.11971432169073622223380591060, 12.47812472166805208031198904371, 13.384034535091711221208928653092, 13.98716045443683845635918934328, 14.71210203442548964863476904238, 15.70988781886096274532114420007, 16.11776149515810781100171150306, 16.67523752104070827278401081185, 17.159750549541029088670849260464, 17.65848853183043693926289645776, 18.70272592393794825012091817538

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