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Properties

Label 1-2001-2001.275-r1-0-0
Degree $1$
Conductor $2001$
Sign $-0.592 - 0.805i$
Analytic cond. $215.037$
Root an. cond. $215.037$
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  + (0.974 − 0.222i)2-s + (0.900 − 0.433i)4-s + (0.222 + 0.974i)5-s + (0.900 + 0.433i)7-s + (0.781 − 0.623i)8-s + (0.433 + 0.900i)10-s + (−0.781 − 0.623i)11-s + (−0.623 + 0.781i)13-s + (0.974 + 0.222i)14-s + (0.623 − 0.781i)16-s − i·17-s + (−0.433 − 0.900i)19-s + (0.623 + 0.781i)20-s + (−0.900 − 0.433i)22-s + (−0.900 + 0.433i)25-s + (−0.433 + 0.900i)26-s + ⋯
L(s)  = 1  + (0.974 − 0.222i)2-s + (0.900 − 0.433i)4-s + (0.222 + 0.974i)5-s + (0.900 + 0.433i)7-s + (0.781 − 0.623i)8-s + (0.433 + 0.900i)10-s + (−0.781 − 0.623i)11-s + (−0.623 + 0.781i)13-s + (0.974 + 0.222i)14-s + (0.623 − 0.781i)16-s − i·17-s + (−0.433 − 0.900i)19-s + (0.623 + 0.781i)20-s + (−0.900 − 0.433i)22-s + (−0.900 + 0.433i)25-s + (−0.433 + 0.900i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.592 - 0.805i$
Analytic conductor: \(215.037\)
Root analytic conductor: \(215.037\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (1:\ ),\ -0.592 - 0.805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.072146541 - 2.121098103i\)
\(L(\frac12)\) \(\approx\) \(1.072146541 - 2.121098103i\)
\(L(1)\) \(\approx\) \(1.781558039 - 0.2412473651i\)
\(L(1)\) \(\approx\) \(1.781558039 - 0.2412473651i\)

Euler product

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

−20.16119456008445350073991692686, −19.820374441996237077973484234563, −18.43265539319444175598458308635, −17.55701919506716037314155136493, −16.98918403680272295597598114237, −16.44749121311940577102815070830, −15.41966326878477027991917282105, −14.87215905314477204724934770324, −14.25494721664074426016250430294, −13.16874666785229630573518127163, −12.86749546798862491461798840864, −12.19036229211244368375643288924, −11.29103004907813141491365093241, −10.449028497442171923149171172732, −9.80399141834827064793685751038, −8.3311893596757873307069132677, −8.013036591527644345852579654738, −7.270294595458382806731810359681, −6.04917747122075008478812647205, −5.44001104225261295662434916659, −4.64691277199867229290492110037, −4.21838150675008437076733478517, −3.031021671032364525276265057502, −1.94337539801508931780412346052, −1.35105820226090501989484628117, 0.23328336160841830463496823796, 1.78719518790440270993743838189, 2.413352711178908638345045416254, 3.069457229906293353342585286279, 4.12101443703182513791850435914, 5.09222583848553014459752420073, 5.51018480684771906029250114960, 6.630923493701879419058950670557, 7.162586480277077440695995856082, 8.01486904581664530417491137905, 9.17191826708145886023435892263, 10.04885963071396304615243593347, 11.0465463354846025675847138470, 11.2388766736659166322292561398, 12.03878053548124156477993311282, 13.02139089025588579281876856147, 13.78221230957981793475181740762, 14.34426682394441052815101996251, 14.91821797387835424681372158584, 15.63167898667373851598859367961, 16.378754367329483942635925751509, 17.37661674599913712480191527025, 18.30353504313782654504058737660, 18.74682785797760369297816750459, 19.576990424915596714690720413069

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