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Properties

Label 1-4033-4033.1093-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.670 - 0.742i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (0.642 + 0.766i)2-s + (0.835 + 0.549i)3-s + (−0.173 + 0.984i)4-s + (−0.918 − 0.396i)5-s + (0.116 + 0.993i)6-s + (−0.993 + 0.116i)7-s + (−0.866 + 0.5i)8-s + (0.396 + 0.918i)9-s + (−0.286 − 0.957i)10-s + (0.286 − 0.957i)11-s + (−0.686 + 0.727i)12-s + (0.802 − 0.597i)13-s + (−0.727 − 0.686i)14-s + (−0.549 − 0.835i)15-s + (−0.939 − 0.342i)16-s + (−0.642 − 0.766i)17-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (0.835 + 0.549i)3-s + (−0.173 + 0.984i)4-s + (−0.918 − 0.396i)5-s + (0.116 + 0.993i)6-s + (−0.993 + 0.116i)7-s + (−0.866 + 0.5i)8-s + (0.396 + 0.918i)9-s + (−0.286 − 0.957i)10-s + (0.286 − 0.957i)11-s + (−0.686 + 0.727i)12-s + (0.802 − 0.597i)13-s + (−0.727 − 0.686i)14-s + (−0.549 − 0.835i)15-s + (−0.939 − 0.342i)16-s + (−0.642 − 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.670 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.670 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.670 - 0.742i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1093, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.670 - 0.742i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5534915503 + 1.245660961i\)
\(L(\frac12)\) \(\approx\) \(-0.5534915503 + 1.245660961i\)
\(L(1)\) \(\approx\) \(1.054413786 + 0.8129683817i\)
\(L(1)\) \(\approx\) \(1.054413786 + 0.8129683817i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.642 + 0.766i)T \)
3 \( 1 + (0.835 + 0.549i)T \)
5 \( 1 + (-0.918 - 0.396i)T \)
7 \( 1 + (-0.993 + 0.116i)T \)
11 \( 1 + (0.286 - 0.957i)T \)
13 \( 1 + (0.802 - 0.597i)T \)
17 \( 1 + (-0.642 - 0.766i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (-0.342 + 0.939i)T \)
29 \( 1 + (0.957 - 0.286i)T \)
31 \( 1 + (0.998 - 0.0581i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (0.597 - 0.802i)T \)
53 \( 1 + (-0.686 + 0.727i)T \)
59 \( 1 + (0.549 + 0.835i)T \)
61 \( 1 + (0.957 - 0.286i)T \)
67 \( 1 + (-0.973 + 0.230i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (-0.973 + 0.230i)T \)
79 \( 1 + (0.957 + 0.286i)T \)
83 \( 1 + (0.396 - 0.918i)T \)
89 \( 1 + (-0.998 - 0.0581i)T \)
97 \( 1 + (-0.549 + 0.835i)T \)
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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

−18.153834384196863883088693042264, −17.46861931356538507279490598694, −16.13877030530573951100014987044, −15.55946324514953189879747353605, −15.05660248033609726217874141312, −14.32739194874013716630172322519, −13.74835693418020943253964913940, −12.92821127676004622055538673047, −12.46400265413712349459605052656, −11.99739426126571313010785887515, −10.978071488934490734210499290421, −10.418788633931958748105569910137, −9.59525322488097994078857612361, −8.83033432351837818387353022544, −8.28904661377344852733310778237, −7.06831218324513567606825226399, −6.590868551189997132682112824820, −6.18306563869244384525869258417, −4.49341153416812973765606472119, −4.158794603232176054133824773660, −3.51264577913902650522533073204, −2.63273330426247417999349796678, −2.133844137110301167707408342931, −1.04095402789372107658844167865, −0.166449943348683688119302302100, 0.94532887968176166466637671640, 2.64636227329035634748927964741, 3.08221547664483195643243424852, 3.9529274504603161356996755543, 4.19123156292115009023070792676, 5.256194886725999590086038403596, 6.050757605628631958608648030978, 6.75202992586889812867793119401, 7.7172155659670257130769043544, 8.244196086186979166704617898854, 8.8202716670926141316139246291, 9.39501243744415676934471753816, 10.44104873808661445898382527607, 11.30958856408596777775048767684, 12.0064353546427023451345197950, 12.90181092264876566056888781610, 13.37747540998486831487344006379, 13.93357389797430269743924958891, 14.77412760140547266858431883224, 15.57644932372037158510557276670, 15.84142443368805776643254696252, 16.27637567346345533590275419405, 16.99404197251202329974158388299, 17.98913087387821695889157103988, 18.97452357438792355558821409200

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