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Properties

Label 1-1004-1004.279-r1-0-0
Degree $1$
Conductor $1004$
Sign $0.151 + 0.988i$
Analytic cond. $107.894$
Root an. cond. $107.894$
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  + (−0.954 + 0.297i)3-s + (−0.637 + 0.770i)5-s + (0.0376 − 0.999i)7-s + (0.823 − 0.567i)9-s + (−0.998 + 0.0502i)11-s + (−0.285 − 0.958i)13-s + (0.379 − 0.925i)15-s + (−0.984 − 0.175i)17-s + (−0.656 − 0.754i)19-s + (0.260 + 0.965i)21-s + (−0.899 + 0.437i)23-s + (−0.187 − 0.982i)25-s + (−0.617 + 0.786i)27-s + (−0.711 + 0.702i)29-s + (−0.356 − 0.934i)31-s + ⋯
L(s)  = 1  + (−0.954 + 0.297i)3-s + (−0.637 + 0.770i)5-s + (0.0376 − 0.999i)7-s + (0.823 − 0.567i)9-s + (−0.998 + 0.0502i)11-s + (−0.285 − 0.958i)13-s + (0.379 − 0.925i)15-s + (−0.984 − 0.175i)17-s + (−0.656 − 0.754i)19-s + (0.260 + 0.965i)21-s + (−0.899 + 0.437i)23-s + (−0.187 − 0.982i)25-s + (−0.617 + 0.786i)27-s + (−0.711 + 0.702i)29-s + (−0.356 − 0.934i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1004\)    =    \(2^{2} \cdot 251\)
Sign: $0.151 + 0.988i$
Analytic conductor: \(107.894\)
Root analytic conductor: \(107.894\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1004} (279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1004,\ (1:\ ),\ 0.151 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02055667221 + 0.01763826858i\)
\(L(\frac12)\) \(\approx\) \(0.02055667221 + 0.01763826858i\)
\(L(1)\) \(\approx\) \(0.4723555325 - 0.05595270547i\)
\(L(1)\) \(\approx\) \(0.4723555325 - 0.05595270547i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
251 \( 1 \)
good3 \( 1 + (-0.954 + 0.297i)T \)
5 \( 1 + (-0.637 + 0.770i)T \)
7 \( 1 + (0.0376 - 0.999i)T \)
11 \( 1 + (-0.998 + 0.0502i)T \)
13 \( 1 + (-0.285 - 0.958i)T \)
17 \( 1 + (-0.984 - 0.175i)T \)
19 \( 1 + (-0.656 - 0.754i)T \)
23 \( 1 + (-0.899 + 0.437i)T \)
29 \( 1 + (-0.711 + 0.702i)T \)
31 \( 1 + (-0.356 - 0.934i)T \)
37 \( 1 + (-0.999 - 0.0251i)T \)
41 \( 1 + (-0.997 + 0.0753i)T \)
43 \( 1 + (0.745 - 0.666i)T \)
47 \( 1 + (0.425 - 0.904i)T \)
53 \( 1 + (0.402 - 0.915i)T \)
59 \( 1 + (0.984 - 0.175i)T \)
61 \( 1 + (-0.863 + 0.503i)T \)
67 \( 1 + (0.379 + 0.925i)T \)
71 \( 1 + (-0.793 - 0.607i)T \)
73 \( 1 + (-0.597 - 0.801i)T \)
79 \( 1 + (-0.162 - 0.986i)T \)
83 \( 1 + (-0.979 + 0.199i)T \)
89 \( 1 + (-0.137 - 0.990i)T \)
97 \( 1 + (0.492 + 0.870i)T \)
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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

−22.05575311933541897687883807312, −21.35711983516391284993656590326, −20.65488567195810123457175598962, −19.49343945456489233103764956524, −18.845460797185162984946973754774, −18.22991543161421307358219394253, −17.30100723377176737715256998747, −16.53784569281626396204457973463, −15.812563963235884119547891431963, −15.34433462189115937245990131027, −14.061676603297901795341881572202, −12.93100058049267438774435960713, −12.476684358318352474507787137807, −11.78259630692177307819311038459, −11.06473982142293191491565451708, −10.115176047317313584626544015290, −8.9947583732709565529809912139, −8.27230940279517632209382476733, −7.39120401538736868621746172869, −6.34148181018547846958344431673, −5.55504826070949396303176582378, −4.75216270270626206044016341582, −4.01114042592609719755601587889, −2.36274307573410636688988366928, −1.571331235348426230620936261189, 0.01761268593060320337966308446, 0.3452492703826617934969851390, 2.08855077624521503083857735581, 3.35597994918423868997332255142, 4.15974963971858303536812921435, 5.023327070516539125576016826318, 5.97730778965814280412963280448, 7.11799761951074045301889350256, 7.37420627070525861518505666903, 8.568673683513856066632468289090, 10.004174731177941936159559039906, 10.501793687606246145555359594181, 11.05909651233871206525787317691, 11.84141610324551345254650485687, 12.92651002127771500331817807029, 13.50224164579966684114778623131, 14.79487778227052556273540235312, 15.4507576866729377646223007527, 16.01341771716946569040646344383, 17.005989843000071936262392820724, 17.76405330978890054665778023748, 18.24978718766730567534426398897, 19.27565380112507548582392599444, 20.15102603830642364513763547614, 20.77333956255784058820315836341

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