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Properties

Label 1-4033-4033.54-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.158 + 0.987i$
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 + (−0.766 − 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.642 + 0.766i)5-s + (0.173 + 0.984i)6-s + (0.766 + 0.642i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (0.984 − 0.173i)10-s + (−0.984 − 0.173i)11-s + (0.5 − 0.866i)12-s + (−0.173 + 0.984i)13-s + (−0.173 − 0.984i)14-s + (0.984 − 0.173i)15-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (−0.766 − 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.642 + 0.766i)5-s + (0.173 + 0.984i)6-s + (0.766 + 0.642i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (0.984 − 0.173i)10-s + (−0.984 − 0.173i)11-s + (0.5 − 0.866i)12-s + (−0.173 + 0.984i)13-s + (−0.173 − 0.984i)14-s + (0.984 − 0.173i)15-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.158 + 0.987i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.158 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3833871055 + 0.3266509623i\)
\(L(\frac12)\) \(\approx\) \(0.3833871055 + 0.3266509623i\)
\(L(1)\) \(\approx\) \(0.5153645307 - 0.04519275452i\)
\(L(1)\) \(\approx\) \(0.5153645307 - 0.04519275452i\)

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

−18.14528914372648552167077398405, −17.22788892920542240585599950063, −17.132534819334080215064768071069, −16.41214906985370336550496252282, −15.674440245650603100481730100452, −15.119590243860857802458388501372, −14.748663065900968208810379653660, −13.55691812811303802760174151721, −12.7370973188391256143075208323, −12.030734962139539476756943389635, −11.12423105175319246877157840358, −10.60231823665116893771669700613, −10.16838976226985818851140652276, −9.29955943662451401981722622275, −8.36488420552449431709843042114, −7.93313598168477701509834380345, −7.34311094008793879517220455429, −6.29740880707988698415438505406, −5.39905841154355922466836488393, −5.117427334038890443148145668673, −4.28689972600601139834663304673, −3.5100575469129162438081047171, −2.00647714453105169438506689427, −1.009939497121745475990604248204, −0.30447840572698508886487361920, 0.875412345069791432057578249610, 1.83222120215188417590263616998, 2.66432511173372119893887470413, 3.10013682398173808537742212813, 4.6613986725912018701817350911, 4.86493009373553654615176279894, 6.16211845265334987568782157198, 7.05882488031919064659207931386, 7.343298711345804390164586028795, 8.15647823718122590585026056782, 8.82668155821133624369209266924, 9.702694947228576048973155072, 10.76196831735335628102032125987, 11.09827965661203142372504505696, 11.51557642155626951287304677556, 12.25575362010344645067083932946, 12.81324720400323079579675036501, 13.73678085421893427444339655374, 14.454959533208935268561840252346, 15.52160479863769526130340256830, 16.036580036184977489341946777463, 16.69565031624867860107409335872, 17.64756033290305702753673908654, 18.086829195318087425771019713116, 18.54403697906417433181471932639

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