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Properties

Label 1-4033-4033.46-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.433 + 0.900i$
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 + (0.766 − 0.642i)3-s + (0.766 − 0.642i)4-s + (−0.939 − 0.342i)5-s + (0.5 − 0.866i)6-s + (−0.939 − 0.342i)7-s + (0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s − 10-s − 11-s + (0.173 − 0.984i)12-s + (−0.173 − 0.984i)13-s − 14-s + (−0.939 + 0.342i)15-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.766 − 0.642i)3-s + (0.766 − 0.642i)4-s + (−0.939 − 0.342i)5-s + (0.5 − 0.866i)6-s + (−0.939 − 0.342i)7-s + (0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s − 10-s − 11-s + (0.173 − 0.984i)12-s + (−0.173 − 0.984i)13-s − 14-s + (−0.939 + 0.342i)15-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.433 + 0.900i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.433 + 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.8736348711 - 1.390275776i\)
\(L(\frac12)\) \(\approx\) \(-0.8736348711 - 1.390275776i\)
\(L(1)\) \(\approx\) \(1.044197901 - 1.100375206i\)
\(L(1)\) \(\approx\) \(1.044197901 - 1.100375206i\)

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 \( 1 + (0.766 - 0.642i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.173 - 0.984i)T \)
17 \( 1 + (-0.766 - 0.642i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.766 - 0.642i)T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (-0.939 + 0.342i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
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

−19.01725466266597598840318108412, −18.56886461287140298728235165609, −17.31728212967748100530409138812, −16.291280445925424480984814697589, −16.0338239249501525543786526536, −15.57238892424293124312854069081, −14.87338238532654328477920252052, −14.30463963497963639351537403249, −13.580991231936344499899751379576, −12.7967052175007087780028166729, −12.38218991956120535012396019837, −11.31776229117062425011124425511, −10.81400309707051367042751506026, −10.085630017735341094245580440447, −8.86300000709359852806691788714, −8.67167534194886140356893562791, −7.41761114075101582458179295537, −7.22865604236041874377218735798, −6.28601723301865892451600149713, −5.300710664977841063048109040547, −4.58386703439451795751020269766, −3.977554595377927173106068665282, −3.11571598104932471312809483371, −2.828012276494813229963298991866, −1.88255160898286617479178469551, 0.28672296243412246596925784265, 1.0973198581185193681417550561, 2.32858616523620260718168366175, 2.98590739959091001398379591566, 3.51101757447940912075199223866, 4.214332794447306439596992377, 5.22259528570932142713906447714, 5.87547284540547096024480416277, 6.92652605486003292690784985903, 7.49299266487489048623874055202, 7.87267142101908979681804837648, 9.04201225731786153507783644409, 9.7480777202645571904140676208, 10.46927071955784731559909270892, 11.4629605498184336147132355180, 11.93527382643122045554919944124, 12.83517436711295503804905751312, 13.24454414963700124584500896090, 13.49190677577006559377548685177, 14.62643770786502212336927680538, 15.25395680867962695502737292533, 15.76328470206271737501631576668, 16.22529038648497768474151599851, 17.34520458358709104675738364387, 18.36979533087919949640649473461

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