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Properties

Label 1-3997-3997.580-r1-0-0
Degree $1$
Conductor $3997$
Sign $-0.918 - 0.395i$
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  + (0.851 − 0.523i)2-s + (−0.999 − 0.0220i)3-s + (0.451 − 0.892i)4-s + (0.992 − 0.120i)5-s + (−0.863 + 0.504i)6-s + (−0.0825 − 0.996i)8-s + (0.999 + 0.0440i)9-s + (0.782 − 0.622i)10-s + (0.930 − 0.366i)11-s + (−0.471 + 0.882i)12-s + (−0.0935 − 0.995i)13-s + (−0.995 + 0.0990i)15-s + (−0.592 − 0.805i)16-s + (0.949 − 0.314i)17-s + (0.874 − 0.485i)18-s + (0.795 + 0.605i)19-s + ⋯
L(s)  = 1  + (0.851 − 0.523i)2-s + (−0.999 − 0.0220i)3-s + (0.451 − 0.892i)4-s + (0.992 − 0.120i)5-s + (−0.863 + 0.504i)6-s + (−0.0825 − 0.996i)8-s + (0.999 + 0.0440i)9-s + (0.782 − 0.622i)10-s + (0.930 − 0.366i)11-s + (−0.471 + 0.882i)12-s + (−0.0935 − 0.995i)13-s + (−0.995 + 0.0990i)15-s + (−0.592 − 0.805i)16-s + (0.949 − 0.314i)17-s + (0.874 − 0.485i)18-s + (0.795 + 0.605i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $-0.918 - 0.395i$
Analytic conductor: \(429.537\)
Root analytic conductor: \(429.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (580, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3997,\ (1:\ ),\ -0.918 - 0.395i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9256401896 - 4.492898389i\)
\(L(\frac12)\) \(\approx\) \(0.9256401896 - 4.492898389i\)
\(L(1)\) \(\approx\) \(1.445524416 - 1.092449042i\)
\(L(1)\) \(\approx\) \(1.445524416 - 1.092449042i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
571 \( 1 \)
good2 \( 1 + (0.851 - 0.523i)T \)
3 \( 1 + (-0.999 - 0.0220i)T \)
5 \( 1 + (0.992 - 0.120i)T \)
11 \( 1 + (0.930 - 0.366i)T \)
13 \( 1 + (-0.0935 - 0.995i)T \)
17 \( 1 + (0.949 - 0.314i)T \)
19 \( 1 + (0.795 + 0.605i)T \)
23 \( 1 + (0.652 - 0.757i)T \)
29 \( 1 + (-0.298 - 0.954i)T \)
31 \( 1 + (0.401 - 0.915i)T \)
37 \( 1 + (0.509 + 0.860i)T \)
41 \( 1 + (0.191 - 0.981i)T \)
43 \( 1 + (-0.834 - 0.551i)T \)
47 \( 1 + (-0.993 + 0.110i)T \)
53 \( 1 + (0.761 - 0.648i)T \)
59 \( 1 + (-0.245 - 0.969i)T \)
61 \( 1 + (-0.565 + 0.824i)T \)
67 \( 1 + (-0.942 - 0.335i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (0.999 - 0.0110i)T \)
79 \( 1 + (0.775 + 0.631i)T \)
83 \( 1 + (0.868 + 0.495i)T \)
89 \( 1 + (-0.00551 - 0.999i)T \)
97 \( 1 + (-0.988 - 0.153i)T \)
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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

−18.14507075985128266819583336327, −17.84793538551824271005398711008, −17.031995298552001554502792641342, −16.58086706025824361549062690731, −16.150794692220562853036581660158, −14.982742044214493113353543496129, −14.6290165357312318028163234167, −13.7504576614889618201971520358, −13.27273819556160081643412087783, −12.38513339456481814378850037511, −11.929446126881528527406647658681, −11.22242655877962031314826396944, −10.52034260076965721496460442683, −9.44877510323252199599151671430, −9.16654420264421337852078468085, −7.78424456802913585144372671147, −6.972351487167428315989805880, −6.59942159947701744310028709348, −5.899242737323882145800441124667, −5.142296694555494086296424591351, −4.711129384495676194949343545, −3.7266392248222051718431172687, −2.9281995227019181327057713653, −1.68426345881062419804910306378, −1.24608853860330303931284783677, 0.58999192578368422684238581877, 1.059159560023995319699385559091, 1.908769328805208203248134915262, 2.90830506129128625700177195325, 3.69515525639200894830632064050, 4.62638539415878013108840476771, 5.36405775984831117750313456042, 5.79132937966166185730739839328, 6.40968151173153246195876100899, 7.132812830154187714821188328229, 8.19061260174838891571373675026, 9.54639578771053189247561634021, 9.81070317387015635999295601960, 10.52388504925430601771948226763, 11.236887042142681650417095915153, 11.99147453731256201093806727334, 12.38379036346154768566269830340, 13.29802618599892063249254449647, 13.635826261986766189103858737817, 14.579741328066473925275429900348, 15.09742832016802948755022497127, 16.06183084140620139369510750094, 16.77770936039301015611025250293, 17.14530535892345301554991268732, 18.17204977313833339635435059953

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