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Properties

Label 1-6001-6001.2530-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.972 + 0.232i$
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 + (0.831 + 0.555i)3-s + 4-s + (0.980 − 0.195i)5-s + (0.831 + 0.555i)6-s + (0.195 − 0.980i)7-s + 8-s + (0.382 + 0.923i)9-s + (0.980 − 0.195i)10-s + (0.382 − 0.923i)11-s + (0.831 + 0.555i)12-s + (−0.195 + 0.980i)13-s + (0.195 − 0.980i)14-s + (0.923 + 0.382i)15-s + 16-s + ⋯
L(s)  = 1  + 2-s + (0.831 + 0.555i)3-s + 4-s + (0.980 − 0.195i)5-s + (0.831 + 0.555i)6-s + (0.195 − 0.980i)7-s + 8-s + (0.382 + 0.923i)9-s + (0.980 − 0.195i)10-s + (0.382 − 0.923i)11-s + (0.831 + 0.555i)12-s + (−0.195 + 0.980i)13-s + (0.195 − 0.980i)14-s + (0.923 + 0.382i)15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.972 + 0.232i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (2530, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.972 + 0.232i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(7.231802011 + 0.8506659986i\)
\(L(\frac12)\) \(\approx\) \(7.231802011 + 0.8506659986i\)
\(L(1)\) \(\approx\) \(3.320611682 + 0.2725215849i\)
\(L(1)\) \(\approx\) \(3.320611682 + 0.2725215849i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (0.831 + 0.555i)T \)
5 \( 1 + (0.980 - 0.195i)T \)
7 \( 1 + (0.195 - 0.980i)T \)
11 \( 1 + (0.382 - 0.923i)T \)
13 \( 1 + (-0.195 + 0.980i)T \)
19 \( 1 + (0.923 + 0.382i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (0.382 + 0.923i)T \)
31 \( 1 + (-0.980 + 0.195i)T \)
37 \( 1 + (-0.831 + 0.555i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (-0.382 + 0.923i)T \)
47 \( 1 + (0.923 + 0.382i)T \)
53 \( 1 + (-0.980 + 0.195i)T \)
59 \( 1 + (0.980 - 0.195i)T \)
61 \( 1 + (0.923 + 0.382i)T \)
67 \( 1 + (0.980 + 0.195i)T \)
71 \( 1 + (-0.980 - 0.195i)T \)
73 \( 1 + (-0.923 + 0.382i)T \)
79 \( 1 + (0.980 + 0.195i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.980 + 0.195i)T \)
97 \( 1 + (0.382 - 0.923i)T \)
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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

−17.64725015226184394391619799301, −17.31338787879982485124478783675, −16.00126267987329441736633559354, −15.46224672760413662409361720237, −14.85431351486418388516482204631, −14.45556921353034328051094222477, −13.67209159855056318371423847141, −13.25510561776873933339471241622, −12.51142458989201090034259226599, −12.07047207298231827703753259722, −11.384331036009899964361980624243, −10.25334781341558530235041403014, −9.758212585808337763239329766674, −9.079799218346530532665922855128, −8.17346051795273156059847325448, −7.45293210592443823957347966247, −6.87030310325926223469527581377, −6.094299743596380762574495467944, −5.46569674133574813144862844105, −4.91822912320397931177946754059, −3.75109744958557871430864754753, −3.16476390751287286818769194784, −2.21556383498033783475737308143, −2.109919946328884936543945135117, −1.16755910427387062174165418419, 1.253538263820997168470129590615, 1.75274703895403189803657646738, 2.64120039880479063859079178928, 3.46145219446041765702136710057, 3.921063429682666505296685789495, 4.78988431473820527900618460681, 5.252090285298919610961390981197, 6.198395397610381676617205239188, 6.86707635576476168902485349982, 7.53279982638167318186149518325, 8.44929018762638196247102850615, 9.06738973547379195222651547177, 9.98436953156572013766004933918, 10.37993910153850620900881229784, 11.09733154489488005220748402415, 11.846771730854696918466145987508, 12.75543504723955485618664828359, 13.444321185368515089676400089600, 13.95643833203634413753338636086, 14.33654387927590484656500745516, 14.627324997187400785341982062819, 15.98491468392859134205062110223, 16.22835132406500624480726320369, 16.783072416914047620201807861899, 17.481770577946955777827182198563

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