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Properties

Label 1-2001-2001.602-r1-0-0
Degree $1$
Conductor $2001$
Sign $-0.457 + 0.889i$
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  + (0.262 − 0.965i)2-s + (−0.862 − 0.505i)4-s + (0.768 + 0.639i)5-s + (−0.818 − 0.574i)7-s + (−0.714 + 0.699i)8-s + (0.818 − 0.574i)10-s + (−0.591 + 0.806i)11-s + (−0.0611 + 0.998i)13-s + (−0.768 + 0.639i)14-s + (0.488 + 0.872i)16-s + (−0.142 + 0.989i)17-s + (0.862 + 0.505i)19-s + (−0.339 − 0.940i)20-s + (0.623 + 0.781i)22-s + (0.182 + 0.983i)25-s + (0.947 + 0.320i)26-s + ⋯
L(s)  = 1  + (0.262 − 0.965i)2-s + (−0.862 − 0.505i)4-s + (0.768 + 0.639i)5-s + (−0.818 − 0.574i)7-s + (−0.714 + 0.699i)8-s + (0.818 − 0.574i)10-s + (−0.591 + 0.806i)11-s + (−0.0611 + 0.998i)13-s + (−0.768 + 0.639i)14-s + (0.488 + 0.872i)16-s + (−0.142 + 0.989i)17-s + (0.862 + 0.505i)19-s + (−0.339 − 0.940i)20-s + (0.623 + 0.781i)22-s + (0.182 + 0.983i)25-s + (0.947 + 0.320i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.457 + 0.889i$
Analytic conductor: \(215.037\)
Root analytic conductor: \(215.037\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (602, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (1:\ ),\ -0.457 + 0.889i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4790895054 + 0.7850020295i\)
\(L(\frac12)\) \(\approx\) \(0.4790895054 + 0.7850020295i\)
\(L(1)\) \(\approx\) \(0.9860743444 - 0.1926072905i\)
\(L(1)\) \(\approx\) \(0.9860743444 - 0.1926072905i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.262 - 0.965i)T \)
5 \( 1 + (0.768 + 0.639i)T \)
7 \( 1 + (-0.818 - 0.574i)T \)
11 \( 1 + (-0.591 + 0.806i)T \)
13 \( 1 + (-0.0611 + 0.998i)T \)
17 \( 1 + (-0.142 + 0.989i)T \)
19 \( 1 + (0.862 + 0.505i)T \)
31 \( 1 + (0.992 + 0.122i)T \)
37 \( 1 + (-0.794 - 0.607i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (0.992 - 0.122i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (-0.742 + 0.670i)T \)
59 \( 1 + (0.959 + 0.281i)T \)
61 \( 1 + (-0.101 - 0.994i)T \)
67 \( 1 + (-0.591 - 0.806i)T \)
71 \( 1 + (-0.882 - 0.470i)T \)
73 \( 1 + (0.301 - 0.953i)T \)
79 \( 1 + (-0.488 + 0.872i)T \)
83 \( 1 + (0.452 + 0.891i)T \)
89 \( 1 + (-0.523 + 0.852i)T \)
97 \( 1 + (-0.970 - 0.242i)T \)
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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

−19.31847825580645932339191703753, −18.66482059864384540066067245845, −17.73251494600959264219170760277, −17.49676988621138519757318975329, −16.26889808070754714321361703649, −16.01007407508460657226935626828, −15.43097452643642738711055919359, −14.3070971415971328546665617851, −13.597947008833777849372528828390, −13.14343444565010300827550910778, −12.47728081940266288780885230368, −11.63105915021568836449250256721, −10.29285566042440155416686024148, −9.6328396077270535338244400512, −8.935416993627605626347840541, −8.31227488120904335970522492279, −7.38725567848651311301251210792, −6.49923407338840613468716987554, −5.58952329256048857533056707188, −5.422004800993070206659226054056, −4.41736705420570398889240734218, −3.06805041790654727763297995874, −2.71913641531799453757875965230, −0.92436252939232380253690783494, −0.170370638844058179542812388119, 1.24227791630149291008356078098, 2.038848957591838679367357481227, 2.84335871940756326698357532135, 3.68263786644742569212744048397, 4.46031456660930916148866393548, 5.46079772968215968036695227291, 6.28591136386158954744043156765, 6.99763692040929421547423205568, 8.04235703273648087645690052457, 9.309005725707418731927835374770, 9.67746658195769339280054143030, 10.43791447576542241868656039028, 10.88326407807807267622604908486, 12.00430504933647096395578045741, 12.62202866965865071363525804275, 13.43800205860359207869030567678, 13.90716722539818476145462060920, 14.63468687953462485340135075279, 15.4543350338606329368440029543, 16.47935068072737925897567076255, 17.37644464430890692719119726087, 17.91191715635097119454628360781, 18.748525456528386635510697095737, 19.288432868436275425671566483726, 19.99576356172196426550458066195

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