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Properties

Label 1-2669-2669.2647-r0-0-0
Degree $1$
Conductor $2669$
Sign $0.997 - 0.0757i$
Analytic cond. $12.3947$
Root an. cond. $12.3947$
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  + (−0.707 − 0.707i)2-s + (−0.608 − 0.793i)3-s + i·4-s + (−0.608 − 0.793i)5-s + (−0.130 + 0.991i)6-s + (−0.382 − 0.923i)7-s + (0.707 − 0.707i)8-s + (−0.258 + 0.965i)9-s + (−0.130 + 0.991i)10-s + (0.991 + 0.130i)11-s + (0.793 − 0.608i)12-s + (−0.866 + 0.5i)13-s + (−0.382 + 0.923i)14-s + (−0.258 + 0.965i)15-s − 16-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.608 − 0.793i)3-s + i·4-s + (−0.608 − 0.793i)5-s + (−0.130 + 0.991i)6-s + (−0.382 − 0.923i)7-s + (0.707 − 0.707i)8-s + (−0.258 + 0.965i)9-s + (−0.130 + 0.991i)10-s + (0.991 + 0.130i)11-s + (0.793 − 0.608i)12-s + (−0.866 + 0.5i)13-s + (−0.382 + 0.923i)14-s + (−0.258 + 0.965i)15-s − 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2669 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2669 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2669\)    =    \(17 \cdot 157\)
Sign: $0.997 - 0.0757i$
Analytic conductor: \(12.3947\)
Root analytic conductor: \(12.3947\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2669} (2647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2669,\ (0:\ ),\ 0.997 - 0.0757i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2061235380 + 0.007822699321i\)
\(L(\frac12)\) \(\approx\) \(0.2061235380 + 0.007822699321i\)
\(L(1)\) \(\approx\) \(0.3525551945 - 0.3029676852i\)
\(L(1)\) \(\approx\) \(0.3525551945 - 0.3029676852i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
157 \( 1 \)
good2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-0.608 - 0.793i)T \)
5 \( 1 + (-0.608 - 0.793i)T \)
7 \( 1 + (-0.382 - 0.923i)T \)
11 \( 1 + (0.991 + 0.130i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.258 - 0.965i)T \)
23 \( 1 + (0.923 - 0.382i)T \)
29 \( 1 + (0.382 - 0.923i)T \)
31 \( 1 + (-0.991 + 0.130i)T \)
37 \( 1 + (-0.991 + 0.130i)T \)
41 \( 1 + (0.382 + 0.923i)T \)
43 \( 1 + (-0.965 - 0.258i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.258 - 0.965i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (0.991 + 0.130i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.991 + 0.130i)T \)
73 \( 1 + (-0.608 - 0.793i)T \)
79 \( 1 + (-0.923 + 0.382i)T \)
83 \( 1 + (-0.258 + 0.965i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.991 - 0.130i)T \)
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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

−19.09068170969559184508574361508, −18.63255836017884415230912574323, −17.74532630423921107507312028808, −17.193276001513300670357543091, −16.43425079944539500676096174315, −15.83616183537093791377663265638, −15.16964756058619598152877137124, −14.69754816630695624021512367942, −14.2002115591651669608549395795, −12.63191534604898381074909071479, −11.961968307162340336910745099673, −11.28458910710517492608978602376, −10.49875937612753576236973424599, −9.98623379804387842329874091677, −9.06428428547163247093970532417, −8.69251317468735707825364909265, −7.47317521668741369209308560909, −6.89576659474514113286698167134, −6.06808135470512316123970538215, −5.53129499523356401681526487741, −4.64607249930371509280079045336, −3.6387525561454095751854318909, −2.857533818785542048819714240136, −1.56879117112423579896146259295, −0.13189829616565097912056743155, 0.771889553212158531545386156375, 1.4529279162465871367883604378, 2.43307932436474834792270908019, 3.5484264607373798936240348485, 4.4089097154128977088307485361, 4.95459086451379321258530075704, 6.4289502104006682366177902578, 7.14313131956886569114871839748, 7.45828832584296963528761762709, 8.549889350121492270512030232379, 9.09443451577541038660549750804, 9.991625911661211549827000153648, 10.79418265225095168640934855712, 11.638772124369055484425247438778, 11.89463057494911106823106009804, 12.813625153900556684528468038852, 13.195729791884897318395732632735, 14.031007646459029205026464607615, 15.17483279559267364867567215992, 16.3751380962595226786611506361, 16.64831263451451636832822635622, 17.271243786382414110328167708407, 17.70237923634505048991774509474, 18.87698074827567706075943997678, 19.38169242382087968960778856320

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