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Properties

Label 1-6041-6041.167-r0-0-0
Degree $1$
Conductor $6041$
Sign $0.885 + 0.463i$
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  + (−0.861 − 0.507i)2-s + (−0.987 − 0.159i)3-s + (0.485 + 0.874i)4-s + (−0.0909 + 0.995i)5-s + (0.769 + 0.638i)6-s + (0.0255 − 0.999i)8-s + (0.949 + 0.315i)9-s + (0.583 − 0.812i)10-s + (0.980 − 0.195i)11-s + (−0.339 − 0.940i)12-s + (0.393 + 0.919i)13-s + (0.248 − 0.968i)15-s + (−0.529 + 0.848i)16-s + (−0.899 − 0.436i)17-s + (−0.657 − 0.753i)18-s + (0.926 − 0.376i)19-s + ⋯
L(s)  = 1  + (−0.861 − 0.507i)2-s + (−0.987 − 0.159i)3-s + (0.485 + 0.874i)4-s + (−0.0909 + 0.995i)5-s + (0.769 + 0.638i)6-s + (0.0255 − 0.999i)8-s + (0.949 + 0.315i)9-s + (0.583 − 0.812i)10-s + (0.980 − 0.195i)11-s + (−0.339 − 0.940i)12-s + (0.393 + 0.919i)13-s + (0.248 − 0.968i)15-s + (−0.529 + 0.848i)16-s + (−0.899 − 0.436i)17-s + (−0.657 − 0.753i)18-s + (0.926 − 0.376i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6041\)    =    \(7 \cdot 863\)
Sign: $0.885 + 0.463i$
Analytic conductor: \(28.0542\)
Root analytic conductor: \(28.0542\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6041} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6041,\ (0:\ ),\ 0.885 + 0.463i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7561615928 + 0.1859616229i\)
\(L(\frac12)\) \(\approx\) \(0.7561615928 + 0.1859616229i\)
\(L(1)\) \(\approx\) \(0.5784232329 + 0.008449805603i\)
\(L(1)\) \(\approx\) \(0.5784232329 + 0.008449805603i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
863 \( 1 \)
good2 \( 1 + (-0.861 - 0.507i)T \)
3 \( 1 + (-0.987 - 0.159i)T \)
5 \( 1 + (-0.0909 + 0.995i)T \)
11 \( 1 + (0.980 - 0.195i)T \)
13 \( 1 + (0.393 + 0.919i)T \)
17 \( 1 + (-0.899 - 0.436i)T \)
19 \( 1 + (0.926 - 0.376i)T \)
23 \( 1 + (0.941 - 0.335i)T \)
29 \( 1 + (-0.985 - 0.166i)T \)
31 \( 1 + (0.346 + 0.938i)T \)
37 \( 1 + (-0.00364 - 0.999i)T \)
41 \( 1 + (-0.547 - 0.836i)T \)
43 \( 1 + (-0.838 - 0.544i)T \)
47 \( 1 + (-0.920 + 0.390i)T \)
53 \( 1 + (0.953 - 0.301i)T \)
59 \( 1 + (-0.0837 + 0.996i)T \)
61 \( 1 + (-0.917 - 0.396i)T \)
67 \( 1 + (0.908 + 0.416i)T \)
71 \( 1 + (-0.148 + 0.988i)T \)
73 \( 1 + (0.998 - 0.0582i)T \)
79 \( 1 + (-0.629 - 0.776i)T \)
83 \( 1 + (-0.612 - 0.790i)T \)
89 \( 1 + (0.141 + 0.989i)T \)
97 \( 1 + (0.297 - 0.954i)T \)
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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

−17.29097163713084328471791404704, −17.10683951294187800326276114460, −16.580404091065115092205935904537, −15.84262714091114384936936554, −15.26373052474827414201266366242, −14.83205821010493280031390509832, −13.52682536527963608667266129312, −13.09436128496195415792677473154, −12.190918468157667266805834108639, −11.483938004719541943966999878031, −11.17363474244672764058313962780, −10.183931996536354438402415948402, −9.60922506171449949643345572955, −9.10912306046286915657128190460, −8.25615000560501316664030009309, −7.69012648747625686875967154724, −6.75469752162420698591724727455, −6.26811197605040822289635657065, −5.46239717378787932102636370639, −5.00882385911569490542806044678, −4.19894730360486978780276753483, −3.266983435567289734707497682451, −1.749473594171153312442105188266, −1.29875546616699644461227680875, −0.50128942614409533025660048963, 0.65492708609022583082903795717, 1.58030503392245784316398319022, 2.226895906616992559956680991196, 3.25918205037555127493440599374, 3.887924949349140628061248655376, 4.69761696288749814544813732497, 5.7800278123396300653318505245, 6.6601075481927103547124312453, 6.93608166071081789502330538033, 7.42325183272671812264172290703, 8.61576480142155908019460172922, 9.20015348544796325914955917925, 9.86793433433587738457948955230, 10.67288772353276762319604000112, 11.17041814065223704982636091902, 11.62498771633579082721857505101, 12.04216029350676281499832837827, 13.06012517236023771852493575797, 13.67131560286391801890453725549, 14.47222472246408685729436405803, 15.481635833423365408943899415588, 15.9185612715320525342784924846, 16.71047219384201993904552202602, 17.1637537994043544930392474212, 17.92152005494289227422036874304

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