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Properties

Label 1-2005-2005.133-r0-0-0
Degree $1$
Conductor $2005$
Sign $0.999 + 0.00878i$
Analytic cond. $9.31118$
Root an. cond. $9.31118$
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 + (0.382 − 0.923i)3-s − i·4-s + (0.382 + 0.923i)6-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.707 + 0.707i)11-s + (−0.923 − 0.382i)12-s + (−0.382 − 0.923i)13-s + 14-s − 16-s + (0.382 + 0.923i)17-s + 18-s + (0.382 + 0.923i)19-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.382 − 0.923i)3-s − i·4-s + (0.382 + 0.923i)6-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.707 + 0.707i)11-s + (−0.923 − 0.382i)12-s + (−0.382 − 0.923i)13-s + 14-s − 16-s + (0.382 + 0.923i)17-s + 18-s + (0.382 + 0.923i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2005\)    =    \(5 \cdot 401\)
Sign: $0.999 + 0.00878i$
Analytic conductor: \(9.31118\)
Root analytic conductor: \(9.31118\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2005} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2005,\ (0:\ ),\ 0.999 + 0.00878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.161606956 + 0.005100120305i\)
\(L(\frac12)\) \(\approx\) \(1.161606956 + 0.005100120305i\)
\(L(1)\) \(\approx\) \(0.8427364522 - 0.03635344398i\)
\(L(1)\) \(\approx\) \(0.8427364522 - 0.03635344398i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
401 \( 1 \)
good2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.382 - 0.923i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-0.382 - 0.923i)T \)
17 \( 1 + (0.382 + 0.923i)T \)
19 \( 1 + (0.382 + 0.923i)T \)
23 \( 1 + (0.923 - 0.382i)T \)
29 \( 1 + T \)
31 \( 1 + (0.382 + 0.923i)T \)
37 \( 1 + (0.382 - 0.923i)T \)
41 \( 1 - T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (0.923 + 0.382i)T \)
59 \( 1 + (-0.923 + 0.382i)T \)
61 \( 1 + (0.382 + 0.923i)T \)
67 \( 1 + (-0.923 - 0.382i)T \)
71 \( 1 + (0.382 - 0.923i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + (0.382 + 0.923i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.707 - 0.707i)T \)
97 \( 1 + (-0.923 + 0.382i)T \)
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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

−19.86683457278420317982098548448, −19.18801257162126867646493933605, −18.88090526006243756745124436758, −17.82553854662090964840934089814, −16.77404489483113836483718191811, −16.5757067826281493105783349384, −15.6883442867210800182120025033, −15.05600002628251651138516291784, −13.83157591849750482909802648522, −13.52335463088782431804604349250, −12.26010844046664889004002897602, −11.57512940321268107494701680080, −11.17352214966063775114316881387, −9.941109882911359090339685855136, −9.58689495075060968356106902303, −8.923505992768039982130404218664, −8.41388967737817319552313063240, −7.22688634175236712831675310996, −6.45386037543671093186299386441, −5.205241451611504675936410988452, −4.42997966215257125291616482609, −3.37521364118312693150938763147, −2.94081588556650444208695898091, −2.079977850532982802456261139456, −0.66377132794294531617315717689, 0.871333721663397001410430780321, 1.46377842254697541911362869821, 2.675661480838573269701138293711, 3.62652280778402541908175145944, 4.78077149559130294327621819131, 5.92738497587430154792273122239, 6.474433349398375117924858741233, 7.24240283010053675081588010, 7.76640430579080242329526631293, 8.56187032882115199619653513584, 9.385532779080462151277975756091, 10.140339704583866222677929664561, 10.73831779496786663163579246858, 12.04741233940472137758602717514, 12.62890805091926521257850302204, 13.425900547699641454647991590486, 14.32382140141510252756496602090, 14.71467180803876284778326617653, 15.56030643363158094460504261720, 16.55408877229040994690303169808, 17.136639208030846414640599770312, 17.71835523266401974363203935342, 18.39551722356422568769597582994, 19.37022876055901786300350251006, 19.632955924030013177006023509939

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