Properties

Label 1-6001-6001.81-r0-0-0
Degree $1$
Conductor $6001$
Sign $-0.980 + 0.198i$
Analytic cond. $27.8685$
Root an. cond. $27.8685$
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  + (−0.654 + 0.755i)2-s + (−0.0713 + 0.997i)3-s + (−0.142 − 0.989i)4-s + (0.0713 − 0.997i)5-s + (−0.707 − 0.707i)6-s + (0.707 − 0.707i)7-s + (0.841 + 0.540i)8-s + (−0.989 − 0.142i)9-s + (0.707 + 0.707i)10-s + (−0.909 + 0.415i)11-s + (0.997 − 0.0713i)12-s + (0.479 − 0.877i)13-s + (0.0713 + 0.997i)14-s + (0.989 + 0.142i)15-s + (−0.959 + 0.281i)16-s + ⋯
L(s)  = 1  + (−0.654 + 0.755i)2-s + (−0.0713 + 0.997i)3-s + (−0.142 − 0.989i)4-s + (0.0713 − 0.997i)5-s + (−0.707 − 0.707i)6-s + (0.707 − 0.707i)7-s + (0.841 + 0.540i)8-s + (−0.989 − 0.142i)9-s + (0.707 + 0.707i)10-s + (−0.909 + 0.415i)11-s + (0.997 − 0.0713i)12-s + (0.479 − 0.877i)13-s + (0.0713 + 0.997i)14-s + (0.989 + 0.142i)15-s + (−0.959 + 0.281i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $-0.980 + 0.198i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ -0.980 + 0.198i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04149528513 + 0.4143399966i\)
\(L(\frac12)\) \(\approx\) \(0.04149528513 + 0.4143399966i\)
\(L(1)\) \(\approx\) \(0.6209407032 + 0.2408472521i\)
\(L(1)\) \(\approx\) \(0.6209407032 + 0.2408472521i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (-0.654 + 0.755i)T \)
3 \( 1 + (-0.0713 + 0.997i)T \)
5 \( 1 + (0.0713 - 0.997i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (-0.909 + 0.415i)T \)
13 \( 1 + (0.479 - 0.877i)T \)
19 \( 1 + (-0.281 + 0.959i)T \)
23 \( 1 + (0.415 - 0.909i)T \)
29 \( 1 + (0.989 + 0.142i)T \)
31 \( 1 + (-0.349 + 0.936i)T \)
37 \( 1 + (-0.977 + 0.212i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (0.909 - 0.415i)T \)
47 \( 1 + (-0.989 + 0.142i)T \)
53 \( 1 + (-0.0713 + 0.997i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.281 - 0.959i)T \)
67 \( 1 + (-0.707 + 0.707i)T \)
71 \( 1 + (-0.800 - 0.599i)T \)
73 \( 1 + (-0.989 + 0.142i)T \)
79 \( 1 + (0.0713 + 0.997i)T \)
83 \( 1 + (0.841 - 0.540i)T \)
89 \( 1 + (-0.212 + 0.977i)T \)
97 \( 1 + (0.540 + 0.841i)T \)
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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

−17.821395325735387249008616851242, −17.18287478352330807928201238934, −16.16913601903281436656675025720, −15.52267361282631256960117847944, −14.6638361333068361120617608308, −13.86594650592776581600960092537, −13.42974045576643382895781186013, −12.76630458188407636745461055421, −11.71890203293408944835002142460, −11.607941339440316487979299090125, −10.88422085024859924270892609600, −10.39717996524729031034672433969, −9.17176143470118552576591674552, −8.84926795247157007053780887702, −7.952192915752382930389432872239, −7.51149000330268915196842018371, −6.803159728133215653368514718, −6.04614649277938083442808982034, −5.24592943101530324645056851118, −4.23947588155023495574057625343, −3.15329581885063584431506471364, −2.66674048278224701373041201516, −2.02292388808106215838585083525, −1.39351083987223966033682925696, −0.15235357187155193279760517153, 0.90923667152799796656790582039, 1.65466801630501790627633285789, 2.82074374140613810145606046598, 3.92157929017435545290888279162, 4.73339306039287017724656639848, 4.99294667341677385124218684240, 5.71836391789323091531535408822, 6.47379910884254586795641635833, 7.59346779402080874189910114977, 8.15018341748320384026971897437, 8.54388225082742456259300762629, 9.285510030113675236533050841517, 10.15160575846627429811177571681, 10.513433592382862711560747197635, 10.965038052044779411064213607201, 12.07507263897345966857879977694, 12.82510570204940894524842434802, 13.682314300497152683705938142593, 14.29189242226011376336656142345, 14.936226787068053035505608991008, 15.60422501621102624644049010305, 16.19714796646876490718594215138, 16.53891516725128622170953788375, 17.4875798092227869923903002246, 17.57276035073635492639953492033

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