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Properties

Label 1-2001-2001.935-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.948 + 0.317i$
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  + (−0.986 + 0.162i)2-s + (0.947 − 0.320i)4-s + (−0.301 − 0.953i)5-s + (0.452 − 0.891i)7-s + (−0.882 + 0.470i)8-s + (0.452 + 0.891i)10-s + (0.488 + 0.872i)11-s + (−0.979 − 0.202i)13-s + (−0.301 + 0.953i)14-s + (0.794 − 0.607i)16-s + (0.841 − 0.540i)17-s + (−0.947 + 0.320i)19-s + (−0.591 − 0.806i)20-s + (−0.623 − 0.781i)22-s + (−0.818 + 0.574i)25-s + (0.999 + 0.0407i)26-s + ⋯
L(s)  = 1  + (−0.986 + 0.162i)2-s + (0.947 − 0.320i)4-s + (−0.301 − 0.953i)5-s + (0.452 − 0.891i)7-s + (−0.882 + 0.470i)8-s + (0.452 + 0.891i)10-s + (0.488 + 0.872i)11-s + (−0.979 − 0.202i)13-s + (−0.301 + 0.953i)14-s + (0.794 − 0.607i)16-s + (0.841 − 0.540i)17-s + (−0.947 + 0.320i)19-s + (−0.591 − 0.806i)20-s + (−0.623 − 0.781i)22-s + (−0.818 + 0.574i)25-s + (0.999 + 0.0407i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.948 + 0.317i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (935, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.948 + 0.317i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02889094512 - 0.1770068457i\)
\(L(\frac12)\) \(\approx\) \(0.02889094512 - 0.1770068457i\)
\(L(1)\) \(\approx\) \(0.5597179274 - 0.1283762602i\)
\(L(1)\) \(\approx\) \(0.5597179274 - 0.1283762602i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.986 + 0.162i)T \)
5 \( 1 + (-0.301 - 0.953i)T \)
7 \( 1 + (0.452 - 0.891i)T \)
11 \( 1 + (0.488 + 0.872i)T \)
13 \( 1 + (-0.979 - 0.202i)T \)
17 \( 1 + (0.841 - 0.540i)T \)
19 \( 1 + (-0.947 + 0.320i)T \)
31 \( 1 + (0.917 + 0.396i)T \)
37 \( 1 + (-0.996 - 0.0815i)T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (-0.917 + 0.396i)T \)
47 \( 1 + (0.222 + 0.974i)T \)
53 \( 1 + (-0.768 - 0.639i)T \)
59 \( 1 + (-0.415 + 0.909i)T \)
61 \( 1 + (-0.182 + 0.983i)T \)
67 \( 1 + (-0.488 + 0.872i)T \)
71 \( 1 + (0.0611 - 0.998i)T \)
73 \( 1 + (-0.523 - 0.852i)T \)
79 \( 1 + (-0.794 - 0.607i)T \)
83 \( 1 + (-0.862 - 0.505i)T \)
89 \( 1 + (0.262 - 0.965i)T \)
97 \( 1 + (-0.685 - 0.728i)T \)
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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

−20.047986342726945301311338034025, −19.36094125744971031464773742295, −18.803992035298460149309076268755, −18.50010875969950407506480701267, −17.27670463506398365723626517548, −17.11691957390483062808886186105, −15.99413496102549726115656481015, −15.23711236217359545810329796056, −14.75666542244763245142444054390, −14.01063503673595499214736635968, −12.665919912578591713078398548069, −11.91117751531635511168956645157, −11.46239244819775579498144552574, −10.6693374828766894666859507449, −9.97563622798620363827514882542, −9.141731522630615205624371511083, −8.27832204621548515026803342569, −7.8529131995812116534948715090, −6.732572645215594395226087648386, −6.28821152451203852954869177781, −5.269967344438207438888594512617, −3.94420340061768469854947063825, −3.02230061147297384573139812709, −2.38095116266284643681312631047, −1.44813765338575120102149768019, 0.0871093092804390280710060328, 1.233153557600492161091630228545, 1.85960828129862984380028302187, 3.12102104381036033940756536149, 4.33856175296057351483505104560, 4.916263134299566313735613350875, 5.947123502536942549323163077816, 7.11809621587209261860899002929, 7.469448955946733280115900895728, 8.30184101821155840800157475773, 9.00516426211399354667434033457, 9.951947298800740942346134941344, 10.2793865688674074333520934514, 11.398055205752469151393513912423, 12.13112926622276333750427009279, 12.58505128009269003902823351050, 13.824725703574245232197926028314, 14.63877897893982441614647246813, 15.24030726802244328357935533167, 16.21406983296975981354712557665, 16.765320075748989744330061573220, 17.42274585161834293026029454814, 17.72060935954221278065913706430, 19.03801156398017060930044870887, 19.49672081458791440888722233272

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