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Properties

Label 1-6041-6041.251-r0-0-0
Degree $1$
Conductor $6041$
Sign $-0.851 + 0.524i$
Analytic cond. $28.0542$
Root an. cond. $28.0542$
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  + (−0.220 + 0.975i)2-s + (0.988 + 0.152i)3-s + (−0.902 − 0.430i)4-s + (−0.553 + 0.832i)5-s + (−0.366 + 0.930i)6-s + (0.618 − 0.785i)8-s + (0.953 + 0.301i)9-s + (−0.690 − 0.723i)10-s + (0.971 + 0.238i)11-s + (−0.826 − 0.562i)12-s + (0.999 − 0.0291i)13-s + (−0.674 + 0.738i)15-s + (0.629 + 0.776i)16-s + (−0.989 − 0.145i)17-s + (−0.504 + 0.863i)18-s + (0.974 − 0.224i)19-s + ⋯
L(s)  = 1  + (−0.220 + 0.975i)2-s + (0.988 + 0.152i)3-s + (−0.902 − 0.430i)4-s + (−0.553 + 0.832i)5-s + (−0.366 + 0.930i)6-s + (0.618 − 0.785i)8-s + (0.953 + 0.301i)9-s + (−0.690 − 0.723i)10-s + (0.971 + 0.238i)11-s + (−0.826 − 0.562i)12-s + (0.999 − 0.0291i)13-s + (−0.674 + 0.738i)15-s + (0.629 + 0.776i)16-s + (−0.989 − 0.145i)17-s + (−0.504 + 0.863i)18-s + (0.974 − 0.224i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6041\)    =    \(7 \cdot 863\)
Sign: $-0.851 + 0.524i$
Analytic conductor: \(28.0542\)
Root analytic conductor: \(28.0542\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6041} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6041,\ (0:\ ),\ -0.851 + 0.524i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6058524858 + 2.138536160i\)
\(L(\frac12)\) \(\approx\) \(0.6058524858 + 2.138536160i\)
\(L(1)\) \(\approx\) \(0.9639428630 + 0.8469709274i\)
\(L(1)\) \(\approx\) \(0.9639428630 + 0.8469709274i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
863 \( 1 \)
good2 \( 1 + (-0.220 + 0.975i)T \)
3 \( 1 + (0.988 + 0.152i)T \)
5 \( 1 + (-0.553 + 0.832i)T \)
11 \( 1 + (0.971 + 0.238i)T \)
13 \( 1 + (0.999 - 0.0291i)T \)
17 \( 1 + (-0.989 - 0.145i)T \)
19 \( 1 + (0.974 - 0.224i)T \)
23 \( 1 + (-0.241 + 0.970i)T \)
29 \( 1 + (0.850 - 0.526i)T \)
31 \( 1 + (0.262 + 0.964i)T \)
37 \( 1 + (-0.346 + 0.938i)T \)
41 \( 1 + (0.332 + 0.943i)T \)
43 \( 1 + (-0.769 - 0.638i)T \)
47 \( 1 + (-0.373 - 0.927i)T \)
53 \( 1 + (-0.148 - 0.988i)T \)
59 \( 1 + (-0.961 + 0.273i)T \)
61 \( 1 + (0.318 + 0.947i)T \)
67 \( 1 + (-0.641 + 0.767i)T \)
71 \( 1 + (-0.936 + 0.349i)T \)
73 \( 1 + (0.809 - 0.586i)T \)
79 \( 1 + (0.134 - 0.990i)T \)
83 \( 1 + (-0.914 + 0.403i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (-0.876 - 0.482i)T \)
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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

−17.64655759504162329511417201006, −16.807696621901287948694695698192, −16.09903408143134559918285967834, −15.562558890427622057808669329606, −14.59656724806189595002673335266, −13.87192396084196434731233892450, −13.57520184217596852655203607723, −12.56580228597331601023047272339, −12.42015347039887400251748130091, −11.45682838510609435758457410127, −10.95946314648056984255538193420, −10.018439653125587203505922482058, −9.16900959306827017703472527529, −8.96484759004267978580451079870, −8.27210790262506087304483614735, −7.76534193605974402593323004794, −6.80354262087858121337435915240, −5.886246765157303776529422149615, −4.65958931783504265338552363122, −4.26250230880722504208518957388, −3.58593899409333564067635394631, −2.990218789396208858388165198066, −1.950767640056521458646565183732, −1.34410106390323974253736819293, −0.61659465030476879186997696960, 1.05368397409413400808033244213, 1.83898162627496451596496024045, 3.09115574303578952148811019354, 3.558623610338161307930397193413, 4.29807651453760455912744046480, 4.93319426041021656431554258952, 6.12624413965873323896534562935, 6.72462685074973757366785528896, 7.19652604423958175577947875314, 7.94002530214834319305578277578, 8.59710361171956630727338953566, 9.05630356344911892993179944343, 9.93398484062645296100545270317, 10.34550417431678883739279509156, 11.398028294836002909938791369537, 11.95123721758727846920059275194, 13.20016810725436148644332240261, 13.64927976388637595877477001939, 14.12658750861712844821011710216, 14.85182575689429612780419955575, 15.36419297920718013070249297581, 15.85198453749132778905111604869, 16.34205982067818339495680929840, 17.45051698133847926637695050857, 18.03342775832030525569711653858

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