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Properties

Label 1-4033-4033.3305-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.988 + 0.147i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
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 + (0.893 + 0.448i)3-s + 4-s + (−0.993 − 0.116i)5-s + (0.893 + 0.448i)6-s + (0.597 − 0.802i)7-s + 8-s + (0.597 + 0.802i)9-s + (−0.993 − 0.116i)10-s + (−0.993 + 0.116i)11-s + (0.893 + 0.448i)12-s + (−0.993 − 0.116i)13-s + (0.597 − 0.802i)14-s + (−0.835 − 0.549i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + 2-s + (0.893 + 0.448i)3-s + 4-s + (−0.993 − 0.116i)5-s + (0.893 + 0.448i)6-s + (0.597 − 0.802i)7-s + 8-s + (0.597 + 0.802i)9-s + (−0.993 − 0.116i)10-s + (−0.993 + 0.116i)11-s + (0.893 + 0.448i)12-s + (−0.993 − 0.116i)13-s + (0.597 − 0.802i)14-s + (−0.835 − 0.549i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.988 + 0.147i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.988 + 0.147i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.367854679 + 0.3248358642i\)
\(L(\frac12)\) \(\approx\) \(4.367854679 + 0.3248358642i\)
\(L(1)\) \(\approx\) \(2.361528596 + 0.1595135400i\)
\(L(1)\) \(\approx\) \(2.361528596 + 0.1595135400i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (0.893 + 0.448i)T \)
5 \( 1 + (-0.993 - 0.116i)T \)
7 \( 1 + (0.597 - 0.802i)T \)
11 \( 1 + (-0.993 + 0.116i)T \)
13 \( 1 + (-0.993 - 0.116i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.597 - 0.802i)T \)
31 \( 1 + (0.973 + 0.230i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (0.893 - 0.448i)T \)
53 \( 1 + (-0.0581 - 0.998i)T \)
59 \( 1 + (-0.835 - 0.549i)T \)
61 \( 1 + (-0.286 + 0.957i)T \)
67 \( 1 + (0.893 - 0.448i)T \)
71 \( 1 + T \)
73 \( 1 + (0.597 + 0.802i)T \)
79 \( 1 + (-0.835 - 0.549i)T \)
83 \( 1 + (-0.686 - 0.727i)T \)
89 \( 1 + (0.396 + 0.918i)T \)
97 \( 1 + (0.973 - 0.230i)T \)
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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

−18.80189411790669562869785041989, −18.01379909330461495927087700005, −16.92146065537988871795811413318, −15.98955169212348929572702648444, −15.44477112427069618694770546434, −15.08898084452963670197868726924, −14.21152236850023051260630775114, −13.98274180059539539065933560118, −12.79127104658245542642514529193, −12.42296336677678958950287800263, −11.90846040702871207659137780936, −11.098173901807747781818136356973, −10.36494060589254274336967556992, −9.32317669960366044761478078784, −8.41235489527351219561969082938, −7.86417586063396971981311323357, −7.3121605700948062656706312896, −6.632892757073092919683558587062, −5.5941380102229564339573641824, −4.74108251810680931279830026718, −4.32933806135436511527295887821, −3.125154756656535766768649798003, −2.74723831645721406291005589568, −2.12335860230806910195891765034, −0.93968977716959423072512332364, 0.90714327895085680364014318828, 2.1364283838335350221867812723, 2.77441121069401828482094175304, 3.51593948404445557581397844440, 4.38019581602609793884697026543, 4.66663800221996831527708486664, 5.347759062304727125536378043326, 6.76507147940932926805511363727, 7.36757787086694497258026604460, 7.89304249914141530627644737770, 8.439807578646089233142096551038, 9.60272584194390205968917846596, 10.46633075338624370856429907160, 10.91147405053862291492607047215, 11.61013695583426681002530872588, 12.545771073667698185459815393983, 13.141088175826727253890548784861, 13.721599920780734087728103418618, 14.476712330771743643276743608954, 15.11304561326548039489750978109, 15.53733456087197172973560751921, 16.020898804976989904209057375108, 17.024399533532738985856528747888, 17.54402979952358910984239257003, 18.8853540838029845219461627036

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