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Properties

Label 1-2672-2672.997-r0-0-0
Degree $1$
Conductor $2672$
Sign $-0.951 - 0.307i$
Analytic cond. $12.4087$
Root an. cond. $12.4087$
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  + (−0.404 + 0.914i)3-s + (0.0378 − 0.999i)5-s + (0.243 + 0.969i)7-s + (−0.672 − 0.739i)9-s + (−0.872 + 0.489i)11-s + (0.686 − 0.726i)13-s + (0.898 + 0.438i)15-s + (0.421 − 0.906i)17-s + (0.811 + 0.584i)19-s + (−0.985 − 0.169i)21-s + (−0.822 − 0.569i)23-s + (−0.997 − 0.0756i)25-s + (0.948 − 0.316i)27-s + (−0.569 − 0.822i)29-s + (−0.965 − 0.261i)31-s + ⋯
L(s)  = 1  + (−0.404 + 0.914i)3-s + (0.0378 − 0.999i)5-s + (0.243 + 0.969i)7-s + (−0.672 − 0.739i)9-s + (−0.872 + 0.489i)11-s + (0.686 − 0.726i)13-s + (0.898 + 0.438i)15-s + (0.421 − 0.906i)17-s + (0.811 + 0.584i)19-s + (−0.985 − 0.169i)21-s + (−0.822 − 0.569i)23-s + (−0.997 − 0.0756i)25-s + (0.948 − 0.316i)27-s + (−0.569 − 0.822i)29-s + (−0.965 − 0.261i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2672\)    =    \(2^{4} \cdot 167\)
Sign: $-0.951 - 0.307i$
Analytic conductor: \(12.4087\)
Root analytic conductor: \(12.4087\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2672} (997, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2672,\ (0:\ ),\ -0.951 - 0.307i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009200718533 + 0.05836624343i\)
\(L(\frac12)\) \(\approx\) \(0.009200718533 + 0.05836624343i\)
\(L(1)\) \(\approx\) \(0.7467341271 + 0.1272235816i\)
\(L(1)\) \(\approx\) \(0.7467341271 + 0.1272235816i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
167 \( 1 \)
good3 \( 1 + (-0.404 + 0.914i)T \)
5 \( 1 + (0.0378 - 0.999i)T \)
7 \( 1 + (0.243 + 0.969i)T \)
11 \( 1 + (-0.872 + 0.489i)T \)
13 \( 1 + (0.686 - 0.726i)T \)
17 \( 1 + (0.421 - 0.906i)T \)
19 \( 1 + (0.811 + 0.584i)T \)
23 \( 1 + (-0.822 - 0.569i)T \)
29 \( 1 + (-0.569 - 0.822i)T \)
31 \( 1 + (-0.965 - 0.261i)T \)
37 \( 1 + (0.739 + 0.672i)T \)
41 \( 1 + (-0.862 - 0.505i)T \)
43 \( 1 + (-0.150 + 0.988i)T \)
47 \( 1 + (0.280 - 0.959i)T \)
53 \( 1 + (-0.832 + 0.553i)T \)
59 \( 1 + (-0.906 + 0.421i)T \)
61 \( 1 + (0.599 + 0.800i)T \)
67 \( 1 + (0.0378 + 0.999i)T \)
71 \( 1 + (-0.132 + 0.991i)T \)
73 \( 1 + (-0.974 - 0.225i)T \)
79 \( 1 + (0.776 - 0.629i)T \)
83 \( 1 + (-0.998 - 0.0567i)T \)
89 \( 1 + (-0.898 + 0.438i)T \)
97 \( 1 + (-0.965 + 0.261i)T \)
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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

−18.7085043975390339759003307963, −18.27953495144907520080995652150, −17.71699106370032266581954070, −16.87559951141929862702043546828, −16.248782030298925956541977640818, −15.42081303487204856881669335582, −14.24131096994578110742202994403, −14.04317876093706037071005523867, −13.28546813029455119309036063393, −12.62094025391381352292477393428, −11.50119774630941748241880811505, −11.06886063012763389069702494619, −10.5903523003442383287970115111, −9.670265222561746375250181952501, −8.50506281713866272646423816039, −7.66028719944108501239555599479, −7.33274506056998496464242295686, −6.451560614084562408226836771060, −5.8451605141623900438417945219, −4.988460832173595685391165424885, −3.736028154271389727342702404434, −3.18260159431110239989337662298, −1.99776056790649252013288968897, −1.36867451489979965681509972023, −0.02007324461093378322907716083, 1.23304767056438635335079809883, 2.396697143107519063009217321434, 3.273568829677164675937366830467, 4.25677385148410046772031470775, 4.999432013653259995326086858589, 5.624292200529398040822184664590, 5.971457422651739522751366995594, 7.517416797059866149671305079405, 8.23955786067941753386013393865, 8.864333980001217701112735512907, 9.76565276406920057830861025313, 10.067557976491625962873014260124, 11.22220528671506738579781820493, 11.80928022547157769594926185347, 12.435918298820709893873150291319, 13.17116896435867608017776549416, 14.075194845057985362385767771721, 15.06427107330812444179991965537, 15.51015813103023781023963723281, 16.2345312400718011489423626405, 16.577975131571293529952453182646, 17.666257951888213517870318873720, 18.16223185221698245853951582599, 18.754335831894721369116165556395, 20.18842038209056482551383518236

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