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Properties

Label 1-6041-6041.139-r0-0-0
Degree $1$
Conductor $6041$
Sign $-0.224 + 0.974i$
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  + (0.283 + 0.958i)2-s + (−0.999 + 0.0437i)3-s + (−0.838 + 0.544i)4-s + (−0.985 + 0.166i)5-s + (−0.325 − 0.945i)6-s + (−0.760 − 0.649i)8-s + (0.996 − 0.0873i)9-s + (−0.439 − 0.898i)10-s + (0.869 − 0.494i)11-s + (0.814 − 0.580i)12-s + (0.629 − 0.776i)13-s + (0.977 − 0.209i)15-s + (0.406 − 0.913i)16-s + (0.262 + 0.964i)17-s + (0.366 + 0.930i)18-s + (−0.571 − 0.820i)19-s + ⋯
L(s)  = 1  + (0.283 + 0.958i)2-s + (−0.999 + 0.0437i)3-s + (−0.838 + 0.544i)4-s + (−0.985 + 0.166i)5-s + (−0.325 − 0.945i)6-s + (−0.760 − 0.649i)8-s + (0.996 − 0.0873i)9-s + (−0.439 − 0.898i)10-s + (0.869 − 0.494i)11-s + (0.814 − 0.580i)12-s + (0.629 − 0.776i)13-s + (0.977 − 0.209i)15-s + (0.406 − 0.913i)16-s + (0.262 + 0.964i)17-s + (0.366 + 0.930i)18-s + (−0.571 − 0.820i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6041\)    =    \(7 \cdot 863\)
Sign: $-0.224 + 0.974i$
Analytic conductor: \(28.0542\)
Root analytic conductor: \(28.0542\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6041} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6041,\ (0:\ ),\ -0.224 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4281325064 + 0.5381761125i\)
\(L(\frac12)\) \(\approx\) \(0.4281325064 + 0.5381761125i\)
\(L(1)\) \(\approx\) \(0.5755199741 + 0.3121553251i\)
\(L(1)\) \(\approx\) \(0.5755199741 + 0.3121553251i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
863 \( 1 \)
good2 \( 1 + (0.283 + 0.958i)T \)
3 \( 1 + (-0.999 + 0.0437i)T \)
5 \( 1 + (-0.985 + 0.166i)T \)
11 \( 1 + (0.869 - 0.494i)T \)
13 \( 1 + (0.629 - 0.776i)T \)
17 \( 1 + (0.262 + 0.964i)T \)
19 \( 1 + (-0.571 - 0.820i)T \)
23 \( 1 + (-0.929 - 0.370i)T \)
29 \( 1 + (-0.373 + 0.927i)T \)
31 \( 1 + (-0.865 - 0.501i)T \)
37 \( 1 + (-0.541 + 0.840i)T \)
41 \( 1 + (-0.127 - 0.991i)T \)
43 \( 1 + (0.765 + 0.644i)T \)
47 \( 1 + (-0.994 - 0.109i)T \)
53 \( 1 + (-0.589 - 0.807i)T \)
59 \( 1 + (-0.559 - 0.828i)T \)
61 \( 1 + (-0.311 + 0.950i)T \)
67 \( 1 + (-0.0255 + 0.999i)T \)
71 \( 1 + (0.994 + 0.101i)T \)
73 \( 1 + (-0.963 + 0.266i)T \)
79 \( 1 + (-0.184 + 0.982i)T \)
83 \( 1 + (-0.105 - 0.994i)T \)
89 \( 1 + (-0.318 + 0.947i)T \)
97 \( 1 + (-0.359 - 0.933i)T \)
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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

−17.61178271800589773671639959363, −16.88352885167680197683432881868, −16.25588014353056031287935569544, −15.64345101902574579217792574587, −14.81050894205710520789470249290, −14.15128172958228222441583218113, −13.43478813896863189913997596254, −12.47375835098429038373830172353, −12.20023559623883157871167744554, −11.63251312221747301180877034556, −11.11371556271520489899316067749, −10.516238169694475607384805878923, −9.55407511776374157576230690420, −9.20467848058446670874652849679, −8.15988552982535997488546834990, −7.421669560825107898645510955353, −6.559563817582663797621052944345, −5.90684327992605771604553675124, −5.09174408927496197383300124471, −4.239551757965609540530769658180, −4.05306149265314467121456749890, −3.22167593082342348505407013821, −1.86410268799147417237437926733, −1.45202352904220679199602888249, −0.36482381966971408250858963581, 0.527450197464765652214919758448, 1.51050723160484315421230723709, 3.097475493811770036014328517885, 3.85708401647170057577034842576, 4.17887054319087812759212186414, 5.12360583606887176117404618428, 5.78251840566223183270822600728, 6.460591578927415120389724898375, 6.9050560357878182197378601377, 7.76145463800463403607048357060, 8.370432638309817298991962239196, 8.9830182104845218206461309613, 9.9829328813833233039104006231, 10.80161067788633408794197237983, 11.30019138667894003967960901047, 12.08331344331847073629266360040, 12.688664954871243043828327229496, 13.13752583023140545403968083837, 14.15371720925637245968892662724, 14.86129778065374853620701405739, 15.32011489481131955151248890073, 16.11560517904431877767759083368, 16.35515831107077912548898007279, 17.20522840936278925183689031788, 17.615578048363153018909142108403

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