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Properties

Label 1-4033-4033.2106-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.240 - 0.970i$
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  + (−0.939 + 0.342i)2-s + (−0.686 − 0.727i)3-s + (0.766 − 0.642i)4-s + (−0.0581 − 0.998i)5-s + (0.893 + 0.448i)6-s + (0.893 + 0.448i)7-s + (−0.5 + 0.866i)8-s + (−0.0581 + 0.998i)9-s + (0.396 + 0.918i)10-s + (0.396 − 0.918i)11-s + (−0.993 − 0.116i)12-s + (−0.835 + 0.549i)13-s + (−0.993 − 0.116i)14-s + (−0.686 + 0.727i)15-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (−0.686 − 0.727i)3-s + (0.766 − 0.642i)4-s + (−0.0581 − 0.998i)5-s + (0.893 + 0.448i)6-s + (0.893 + 0.448i)7-s + (−0.5 + 0.866i)8-s + (−0.0581 + 0.998i)9-s + (0.396 + 0.918i)10-s + (0.396 − 0.918i)11-s + (−0.993 − 0.116i)12-s + (−0.835 + 0.549i)13-s + (−0.993 − 0.116i)14-s + (−0.686 + 0.727i)15-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.240 - 0.970i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.240 - 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3850371288 - 0.4921820527i\)
\(L(\frac12)\) \(\approx\) \(0.3850371288 - 0.4921820527i\)
\(L(1)\) \(\approx\) \(0.5487194033 - 0.1433188298i\)
\(L(1)\) \(\approx\) \(0.5487194033 - 0.1433188298i\)

Euler product

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

−18.43030708923153531588657370086, −17.72025297468122283257699894289, −17.34497198935628423478128265873, −17.188962139644124445612069887613, −15.867564918959050778862748186201, −15.298179364714264459640125211567, −15.009747796520571064305042706058, −14.051921758639805384934953846186, −13.02461387371614575954802852060, −12.03928949055470811790931134114, −11.53994900196323177161801867006, −10.99016862709774553629190069694, −10.47905576306481475668052764405, −9.67583036874759183178762090607, −9.3781818817431110045082468501, −8.14165682767846253610697882202, −7.547442952894413645499022251520, −6.74763379735416906806903098816, −6.34940221854056822113022630751, −5.00678346939131999180127642000, −4.43988519366507606747159852108, −3.58804692869306529080205724878, −2.64403032353318705332853490343, −1.96000524368724529137204634924, −0.76539843336852271746111147934, 0.36157329898355831102431328532, 1.36034791325738641316765938058, 1.827101995646253536972540074713, 2.7029604036953190455827649158, 4.404823513524889915096847965601, 4.92220317946394138722968217073, 5.78216567495577720872182348769, 6.31868786155549823815971130671, 7.09915272174999862446224016159, 7.97270601496956247181552105419, 8.57414940637669222962717952037, 8.83287867584010459723399511705, 9.96310600693407667268278277140, 10.80555737914914791993292451808, 11.35878728206598687807072267369, 12.08593050662872902612157409973, 12.44214617084227405689404268076, 13.50253588162969735074854144560, 14.325309129942198940263995198011, 14.86742539553216859059568416002, 16.05934259899318199133141339285, 16.30550490185936339724151585510, 17.20437679953586118824561829273, 17.36273382306546873913133784801, 18.17608648248486471264850263383

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