Properties

Label 1-4033-4033.632-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.999 + 0.0198i$
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  + (0.766 + 0.642i)2-s + (0.396 + 0.918i)3-s + (0.173 + 0.984i)4-s + (0.686 + 0.727i)5-s + (−0.286 + 0.957i)6-s + (0.973 − 0.230i)7-s + (−0.5 + 0.866i)8-s + (−0.686 + 0.727i)9-s + (0.0581 + 0.998i)10-s + (0.0581 − 0.998i)11-s + (−0.835 + 0.549i)12-s + (−0.286 + 0.957i)13-s + (0.893 + 0.448i)14-s + (−0.396 + 0.918i)15-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.396 + 0.918i)3-s + (0.173 + 0.984i)4-s + (0.686 + 0.727i)5-s + (−0.286 + 0.957i)6-s + (0.973 − 0.230i)7-s + (−0.5 + 0.866i)8-s + (−0.686 + 0.727i)9-s + (0.0581 + 0.998i)10-s + (0.0581 − 0.998i)11-s + (−0.835 + 0.549i)12-s + (−0.286 + 0.957i)13-s + (0.893 + 0.448i)14-s + (−0.396 + 0.918i)15-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.999 + 0.0198i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (632, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.999 + 0.0198i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04294292259 + 4.335110569i\)
\(L(\frac12)\) \(\approx\) \(0.04294292259 + 4.335110569i\)
\(L(1)\) \(\approx\) \(1.328237982 + 1.851753785i\)
\(L(1)\) \(\approx\) \(1.328237982 + 1.851753785i\)

Euler product

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

−18.222267571412376947872806714330, −17.63944253039508737709994427183, −17.049826957721840256911485227701, −15.87104435806748133854386155223, −15.07380216001921626557407222009, −14.48960806683195907888982832746, −13.95570025147699047438212624983, −13.28270431632453021299554873337, −12.64880589001399820695029840570, −12.072607034438211728099733807147, −11.70326614211963742825474166031, −10.55110067830406693587981167878, −9.896400840841041527128095085271, −9.14176886489431068165105367571, −8.4629108578780553010318917070, −7.47020896048144603026599509767, −6.99151408944718059896127434959, −5.7529629371953097641502484204, −5.38200683818501174863951827833, −4.76313973445533088485809057306, −3.74364366889814476685816693701, −2.60550463198810557445056683360, −2.32026001211838963420485075123, −1.271569069853958786878886206014, −0.90179857807475941259230564501, 1.54308510060257795698181035246, 2.39876677741475324554658545497, 3.43057650338130428387875679412, 3.59836741982535521708766627532, 4.7948478269614051292625878741, 5.28998938697399365976523448755, 5.91296933980916710065981822642, 6.851074674262805880011258900507, 7.59912120525596090105222427078, 8.27700970515777604067873822656, 9.05275239357681914546005858424, 9.73187968022714975737580460763, 10.680013794172601607426990850969, 11.33860801370616362971719079487, 11.682731369846009701065411038084, 13.00297360080351631420853935028, 13.82941635845996802264670271528, 14.034194845052001571966207639926, 14.82576584759635253341594257013, 15.00132479727468290266638758932, 16.10727813429255800603893330859, 16.74869632990707322673722195647, 17.04652388854400672089366541870, 18.010270707374109651296715821322, 18.66522483687948592188716611237

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