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Properties

Label 1-6001-6001.84-r0-0-0
Degree $1$
Conductor $6001$
Sign $-0.973 - 0.228i$
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  + (−0.841 + 0.540i)2-s + (−0.212 − 0.977i)3-s + (0.415 − 0.909i)4-s + (0.212 + 0.977i)5-s + (0.707 + 0.707i)6-s + (0.707 − 0.707i)7-s + (0.142 + 0.989i)8-s + (−0.909 + 0.415i)9-s + (−0.707 − 0.707i)10-s + (−0.959 + 0.281i)11-s + (−0.977 − 0.212i)12-s + (0.0713 − 0.997i)13-s + (−0.212 + 0.977i)14-s + (0.909 − 0.415i)15-s + (−0.654 − 0.755i)16-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)2-s + (−0.212 − 0.977i)3-s + (0.415 − 0.909i)4-s + (0.212 + 0.977i)5-s + (0.707 + 0.707i)6-s + (0.707 − 0.707i)7-s + (0.142 + 0.989i)8-s + (−0.909 + 0.415i)9-s + (−0.707 − 0.707i)10-s + (−0.959 + 0.281i)11-s + (−0.977 − 0.212i)12-s + (0.0713 − 0.997i)13-s + (−0.212 + 0.977i)14-s + (0.909 − 0.415i)15-s + (−0.654 − 0.755i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $-0.973 - 0.228i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ -0.973 - 0.228i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03037212636 - 0.2623862661i\)
\(L(\frac12)\) \(\approx\) \(0.03037212636 - 0.2623862661i\)
\(L(1)\) \(\approx\) \(0.6164851726 - 0.05526930934i\)
\(L(1)\) \(\approx\) \(0.6164851726 - 0.05526930934i\)

Euler product

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

−17.96547132736154899987597620773, −17.2658176535866088487899036925, −16.89358900954331579265125114155, −16.00511185360388390263844974625, −15.72997813800291739777766431632, −15.11896576789350757354982320062, −13.85943848493895603175774884738, −13.51759488724431642854066579252, −12.30234945114399975801247658401, −12.04689611700110605624906575903, −11.15543306319259115548956884479, −10.937715248223499345620249998330, −9.770656609722613643342459890744, −9.490680586309218207343878992821, −8.79572860290021666430854198341, −8.32918698094217879848729957493, −7.621176453834243883104878277646, −6.55790440433297331317869574602, −5.53754597843038144934447675489, −5.08646631593045721129935352240, −4.3846270992181168648874634142, −3.52577047523237681846484837125, −2.690336526914975018838267253719, −1.92136356665880961368512521124, −1.08669291365433900605120031830, 0.100878938886636650794198998, 1.09830740236107666275877634925, 1.85809688803265196290285439807, 2.58997422479942055180899187232, 3.341006357280518049391102889647, 4.7667424930293503896062944327, 5.52125659184852163818791069243, 5.97427356936362634926140271615, 6.96232642436924542645047922001, 7.32499301131082707645081170496, 7.86366510534505750578401433072, 8.37229134945390371172631389357, 9.39559559250454725394272972765, 10.37302931320056066439385102305, 10.709380457850119663883578678735, 11.086101682420833377259964415428, 12.110581804121893425530763070236, 12.82570616334355984740087307245, 13.72483190245462825573717223728, 14.2261549175017803836459960746, 14.70547354843649916932280853914, 15.48844991126390880549620679340, 16.23112867357803316028906469099, 17.01047263758842791514486792481, 17.691514735080758000925080746865

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