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Properties

Label 1-4033-4033.1483-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.790 - 0.612i$
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 + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (−0.173 − 0.984i)5-s + (−0.173 + 0.984i)6-s + (0.173 + 0.984i)7-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.173 − 0.984i)10-s + (0.173 + 0.984i)11-s + (−0.5 + 0.866i)12-s + (0.939 + 0.342i)13-s + (−0.173 + 0.984i)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 + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (−0.173 − 0.984i)5-s + (−0.173 + 0.984i)6-s + (0.173 + 0.984i)7-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.173 − 0.984i)10-s + (0.173 + 0.984i)11-s + (−0.5 + 0.866i)12-s + (0.939 + 0.342i)13-s + (−0.173 + 0.984i)14-s + (0.939 − 0.342i)15-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.790 - 0.612i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.790 - 0.612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6097762774 + 1.783217283i\)
\(L(\frac12)\) \(\approx\) \(-0.6097762774 + 1.783217283i\)
\(L(1)\) \(\approx\) \(1.185661354 + 1.075329563i\)
\(L(1)\) \(\approx\) \(1.185661354 + 1.075329563i\)

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 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (0.173 + 0.984i)T \)
11 \( 1 + (0.173 + 0.984i)T \)
13 \( 1 + (0.939 + 0.342i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.766 - 0.642i)T \)
31 \( 1 + (-0.766 + 0.642i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.939 - 0.342i)T \)
53 \( 1 + T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−18.32823145223630843379007848708, −17.55583921446758895647016050465, −16.51371117655924886335241285797, −16.01926985991982913156234854429, −14.91809350222672136824534547464, −14.47603659712617828545893701304, −13.858989901853806803161294044053, −13.33233252592649997580334665368, −12.88357241209054321490844912442, −11.666525229217995276088965823246, −11.40629285854966621599169283153, −10.732249512849931013210663827, −10.152723897952212495832436540510, −8.91209015417759429559862893314, −8.004521668072745459093215614810, −7.374012491365853761130805156834, −6.68426190812039681742426773100, −6.134333336890467435797982898114, −5.54039025830575436481017042103, −4.16826186001580114824679879706, −3.65039821225101650440957456705, −3.01927942250377353048521145703, −2.11069062729463288978913306608, −1.39148919996185217172894835631, −0.31173247750023354817138693657, 1.92231323949491095095351873158, 2.11897236578898955893475599017, 3.481996500573638621290408651871, 4.07989696724530354003892733693, 4.62613210180667153295457874097, 5.30708722537324978361178878636, 5.94749703264896519999457979308, 6.71184835999571207595573126397, 7.87246077171053515422653492715, 8.62655300820797874968010059638, 8.87350511380089150420650291069, 9.83950696720161060842852244504, 10.80130083144881255296658231660, 11.54723539539785350772635250668, 11.998399996673951445237948724259, 12.89084880134961833827955903973, 13.318414169186754080898494053540, 14.28732244277666525314447175075, 15.00932929444889068565571148213, 15.42412140327634403035743494894, 15.94467662432951255887180803740, 16.56139141120937047664540749638, 17.33646827568690531118632967004, 17.84380175919813994103908250823, 19.10946473233448422464637039919

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