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Properties

Label 1-4033-4033.2085-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.958 + 0.284i$
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 + (−0.597 − 0.802i)3-s + 4-s + (−0.230 + 0.973i)5-s + (0.597 + 0.802i)6-s + (−0.286 − 0.957i)7-s − 8-s + (−0.286 + 0.957i)9-s + (0.230 − 0.973i)10-s + (−0.230 − 0.973i)11-s + (−0.597 − 0.802i)12-s + (−0.973 − 0.230i)13-s + (0.286 + 0.957i)14-s + (0.918 − 0.396i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (−0.597 − 0.802i)3-s + 4-s + (−0.230 + 0.973i)5-s + (0.597 + 0.802i)6-s + (−0.286 − 0.957i)7-s − 8-s + (−0.286 + 0.957i)9-s + (0.230 − 0.973i)10-s + (−0.230 − 0.973i)11-s + (−0.597 − 0.802i)12-s + (−0.973 − 0.230i)13-s + (0.286 + 0.957i)14-s + (0.918 − 0.396i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.958 + 0.284i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2085, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.958 + 0.284i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02870808827 - 0.1977420880i\)
\(L(\frac12)\) \(\approx\) \(0.02870808827 - 0.1977420880i\)
\(L(1)\) \(\approx\) \(0.4335384601 - 0.1333588108i\)
\(L(1)\) \(\approx\) \(0.4335384601 - 0.1333588108i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.597 - 0.802i)T \)
5 \( 1 + (-0.230 + 0.973i)T \)
7 \( 1 + (-0.286 - 0.957i)T \)
11 \( 1 + (-0.230 - 0.973i)T \)
13 \( 1 + (-0.973 - 0.230i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.957 - 0.286i)T \)
31 \( 1 + (-0.448 + 0.893i)T \)
41 \( 1 + (0.984 + 0.173i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (-0.802 - 0.597i)T \)
53 \( 1 + (0.116 + 0.993i)T \)
59 \( 1 + (-0.396 - 0.918i)T \)
61 \( 1 + (0.549 - 0.835i)T \)
67 \( 1 + (-0.802 - 0.597i)T \)
71 \( 1 - T \)
73 \( 1 + (0.286 - 0.957i)T \)
79 \( 1 + (-0.396 - 0.918i)T \)
83 \( 1 + (0.0581 - 0.998i)T \)
89 \( 1 + (0.727 + 0.686i)T \)
97 \( 1 + (0.448 + 0.893i)T \)
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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

−18.736921917092020894553361984219, −17.91039897355865599900530693011, −17.480348831996898879650500296964, −16.839847027945077713544178705994, −16.12239831548344906493919500115, −15.746982356060846963663838810866, −15.04700974190546212148031190064, −14.55542388408107932613803701991, −12.887903440463939700454017991130, −12.47675583793110069439138945635, −11.85582718915853528680839176383, −11.33730053046318138059358571105, −10.24925002724146958911328951961, −9.88437179966788677233534202070, −9.03123823907861935078068340297, −8.82826239568853528919833802384, −7.80768505006385168264089213121, −6.99274393947887319078099903201, −6.154732173347318774971554506211, −5.41331598974530314394244481846, −4.76640067431485271706785802362, −3.97649855007740842910646808469, −2.77575094538585473112833069694, −2.11583338874876251195556263773, −0.97661171152982779908272228476, 0.12076145175383220219827243477, 0.89850850477660203330463450260, 1.88326958085706536609253102128, 2.99159262841385548695504433766, 3.19656011807831603472224197204, 4.71682290763823543501138943330, 5.70295249324113847662776509554, 6.415347012005328316056614626758, 7.03065291486204421148313626731, 7.634191089815373092201789228982, 7.90626592508807225410668893173, 9.1503367860924845537020237217, 9.95672603357967301763246957830, 10.55793296276103373894539752673, 11.17214569467154876782097057523, 11.647064280710270228610739713788, 12.372519054349044047956042089160, 13.37683374536260880958998135, 13.98886310781127950367967246315, 14.62633907876699559241112813753, 15.847607020530701047266101464846, 16.10839825440179159820476674827, 16.89112096475034640990494538698, 17.75440681345159121431923820117, 17.89290942453350965955711159901

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