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Properties

Label 1-2001-2001.1241-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.951 - 0.306i$
Analytic cond. $9.29260$
Root an. cond. $9.29260$
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  + (0.222 + 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.222 − 0.974i)5-s + (0.900 + 0.433i)7-s + (−0.623 − 0.781i)8-s + (0.900 − 0.433i)10-s + (0.623 − 0.781i)11-s + (0.623 − 0.781i)13-s + (−0.222 + 0.974i)14-s + (0.623 − 0.781i)16-s + 17-s + (0.900 − 0.433i)19-s + (0.623 + 0.781i)20-s + (0.900 + 0.433i)22-s + (−0.900 + 0.433i)25-s + (0.900 + 0.433i)26-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.222 − 0.974i)5-s + (0.900 + 0.433i)7-s + (−0.623 − 0.781i)8-s + (0.900 − 0.433i)10-s + (0.623 − 0.781i)11-s + (0.623 − 0.781i)13-s + (−0.222 + 0.974i)14-s + (0.623 − 0.781i)16-s + 17-s + (0.900 − 0.433i)19-s + (0.623 + 0.781i)20-s + (0.900 + 0.433i)22-s + (−0.900 + 0.433i)25-s + (0.900 + 0.433i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.951 - 0.306i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.951 - 0.306i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.782231624 - 0.2800798443i\)
\(L(\frac12)\) \(\approx\) \(1.782231624 - 0.2800798443i\)
\(L(1)\) \(\approx\) \(1.210765832 + 0.2145676260i\)
\(L(1)\) \(\approx\) \(1.210765832 + 0.2145676260i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.222 + 0.974i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
7 \( 1 + (0.900 + 0.433i)T \)
11 \( 1 + (0.623 - 0.781i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
17 \( 1 + T \)
19 \( 1 + (0.900 - 0.433i)T \)
31 \( 1 + (-0.222 - 0.974i)T \)
37 \( 1 + (-0.623 - 0.781i)T \)
41 \( 1 - T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (-0.222 - 0.974i)T \)
59 \( 1 - T \)
61 \( 1 + (0.900 + 0.433i)T \)
67 \( 1 + (-0.623 - 0.781i)T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + (-0.623 - 0.781i)T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (0.900 - 0.433i)T \)
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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

−20.0856220992079526685804176669, −19.280789976915054602279090420368, −18.56472801756455340043389016999, −18.071726690597726613942601283689, −17.36106887732717888890535066412, −16.45451977508925495659388986991, −15.28787943543113847001595689833, −14.56417186935610042621297918805, −14.130286029478526687624421658433, −13.56263268289747346240323429223, −12.30110147149046221789756265897, −11.76231819737954980924925844210, −11.244939559353754323647335995097, −10.35856945228892260355760091712, −9.89503581172052761909292559191, −8.911780838521673225497967621577, −7.99617493105142453098025024745, −7.197717795763598246532568838945, −6.31179654230820818874392792408, −5.25145078236368020070277000627, −4.43373056617344759914846212228, −3.65706157693789629641810117256, −3.02299377402744494151123201880, −1.723719571066288470054808184196, −1.34738252427122481080539801731, 0.64278789970226860690869990025, 1.50767250481440641872532840207, 3.149639267108273689799999903647, 3.87643013834299007851640819147, 4.81852526113396998714405922475, 5.5555123825042492501361224696, 5.90468667014537353732438070679, 7.24056726488398735201989457507, 7.943286611526225086462233402870, 8.54971056483288862486786527757, 9.07638765543546449429072851245, 10.00227973523862440504186802596, 11.29786176590235784344477892099, 11.89166309617971107004398038146, 12.67784510026220204189443964417, 13.45243100432534144688368059720, 14.12798475834892118427405687345, 14.83382519401250236651904483747, 15.7099162948176728555053654246, 16.120253528186569157240031621, 17.026170590842324210481360045257, 17.45596903344545366551224458413, 18.35225031096404674386247909501, 18.951645202374590216446016250320, 20.002174750380199126924311447975

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