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Properties

Label 1-4033-4033.544-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.621 - 0.783i$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s − 6-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s − 10-s − 11-s + (−0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s − 14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s − 6-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s − 10-s − 11-s + (−0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s − 14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.621 - 0.783i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (544, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.621 - 0.783i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4646798016 - 0.9618431595i\)
\(L(\frac12)\) \(\approx\) \(0.4646798016 - 0.9618431595i\)
\(L(1)\) \(\approx\) \(0.5185040533 - 0.6935297933i\)
\(L(1)\) \(\approx\) \(0.5185040533 - 0.6935297933i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 - T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 + T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + T \)
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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

−18.38303419407923532972733710484, −17.89133286189033522391630577201, −17.40810977501144592821483327909, −16.181909220824732479200297925473, −15.92201956954078385156597114053, −15.3739234027720204692711901042, −15.0094479007244312460818516508, −14.13376068189328364181410968124, −13.3711828829852520792803328378, −12.41665652702500990384524142209, −12.04700461161172240690995575883, −11.1596041666239139296509918729, −10.368618879830392325349888067686, −9.85783501678601377326011161033, −8.793122666677333534334392865619, −8.21882410195151172449825623657, −7.487534709699129277870277731725, −6.50878727656124074639566202407, −5.91613256507291260142130532321, −5.49114771363131390291602974371, −4.55799403076950724106910185859, −3.8065265101989417102566660416, −2.98787034345601479800822215311, −2.67359210040052525238270611263, −0.45219991242051834090234879512, 0.68504570791470627217019702441, 1.196111126603042203709862989, 2.16892147584092023174703259113, 3.09886212187657703192372581268, 3.94295031562348190442664990801, 4.70118405413767095266801802647, 5.32238858391671562020349343812, 6.15037302778163141561078578263, 6.85840602031481359328910729815, 7.91969354701396242949323911388, 8.243583386893582654590140370038, 9.50615490551811852921722996159, 10.00564387189748216359603380823, 10.88477721227545392490986167390, 11.54228663760696715951445366841, 12.137016631824781442018751777813, 12.64099170410726864764775633989, 13.37016105284724715810860809826, 13.82822367546259029245113124473, 14.36143318266162434563304489829, 15.876188875091261640076142711325, 16.071345823451710587292104150130, 16.8286517680694904977871349421, 17.80559540010398055236482608425, 18.33181213529152515318282190506

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