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Properties

Label 1-6001-6001.237-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.938 - 0.345i$
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  − 2-s + (−0.707 + 0.707i)3-s + 4-s + (0.707 − 0.707i)5-s + (0.707 − 0.707i)6-s + (0.707 + 0.707i)7-s − 8-s − i·9-s + (−0.707 + 0.707i)10-s + 11-s + (−0.707 + 0.707i)12-s + (0.707 − 0.707i)13-s + (−0.707 − 0.707i)14-s + i·15-s + 16-s + ⋯
L(s)  = 1  − 2-s + (−0.707 + 0.707i)3-s + 4-s + (0.707 − 0.707i)5-s + (0.707 − 0.707i)6-s + (0.707 + 0.707i)7-s − 8-s − i·9-s + (−0.707 + 0.707i)10-s + 11-s + (−0.707 + 0.707i)12-s + (0.707 − 0.707i)13-s + (−0.707 − 0.707i)14-s + i·15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.938 - 0.345i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.938 - 0.345i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.395879654 - 0.2484813789i\)
\(L(\frac12)\) \(\approx\) \(1.395879654 - 0.2484813789i\)
\(L(1)\) \(\approx\) \(0.8298045408 + 0.02746380661i\)
\(L(1)\) \(\approx\) \(0.8298045408 + 0.02746380661i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + T \)
13 \( 1 + (0.707 - 0.707i)T \)
19 \( 1 + iT \)
23 \( 1 - iT \)
29 \( 1 + T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 - iT \)
43 \( 1 + iT \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + T \)
67 \( 1 + (0.707 - 0.707i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + T \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 - T \)
89 \( 1 + (0.707 - 0.707i)T \)
97 \( 1 - T \)
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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

−17.7522443375793345883184071503, −17.34724710577839964140043710297, −16.77785251702805809301716876301, −16.05517398843887350443224072737, −15.2711468796784363507203417572, −14.27225676377495834987018783469, −13.94810066234282164462803741525, −13.21531660310698468971837501411, −12.21948929132965786297870828031, −11.409024277157414183957901133239, −11.23931627161677885831440828382, −10.5752813201159027351966055907, −9.8386359429996407408482106973, −9.1092202081182281386991828939, −8.37085080821307519014178944952, −7.51899611390524847209328563482, −6.97206169251932906072956239887, −6.48846898408211623589424232008, −5.94262346472555169286107467820, −4.98368812169089437572000094230, −4.02597891775082220135801962357, −2.941576916310961277353473425406, −2.13689463304890425729791362810, −1.29415197425917811329699806529, −1.05880315499073382385632773113, 0.67463278460893340738247352040, 1.29882600931075817269124018554, 2.09371364256845923637050739335, 3.07458185504882813929068823230, 4.05139473110847853182420713981, 4.87937160912385932151405850803, 5.6058697966517317653485321060, 6.243324908446335733633203229713, 6.57819597152484790211714720134, 8.08126410938896434657741534232, 8.37550120769584026092640320481, 9.081843933239863340026214477963, 9.73489970873486958009110389341, 10.22325969241289569616047294949, 10.993299571696764990094385947130, 11.60010556815770771683259005093, 12.25519797346141769612047442069, 12.65950281789399217324408209206, 13.93315366879309732568941275765, 14.66043825883309389331300972633, 15.242908353959117091764548025280, 15.98212088332980360780483015769, 16.49481585464313162613264646442, 17.15624856161435353701366180604, 17.51069910699797510962318232733

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