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Properties

Label 1-4033-4033.1681-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.601 - 0.798i$
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 + (0.173 + 0.984i)3-s + (0.173 + 0.984i)4-s + (0.766 − 0.642i)5-s + (0.5 − 0.866i)6-s + (0.766 − 0.642i)7-s + (0.5 − 0.866i)8-s + (−0.939 + 0.342i)9-s − 10-s − 11-s + (−0.939 + 0.342i)12-s + (0.939 + 0.342i)13-s − 14-s + (0.766 + 0.642i)15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)3-s + (0.173 + 0.984i)4-s + (0.766 − 0.642i)5-s + (0.5 − 0.866i)6-s + (0.766 − 0.642i)7-s + (0.5 − 0.866i)8-s + (−0.939 + 0.342i)9-s − 10-s − 11-s + (−0.939 + 0.342i)12-s + (0.939 + 0.342i)13-s − 14-s + (0.766 + 0.642i)15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.601 - 0.798i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1681, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.601 - 0.798i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.174200050 - 0.5857816747i\)
\(L(\frac12)\) \(\approx\) \(1.174200050 - 0.5857816747i\)
\(L(1)\) \(\approx\) \(0.8934186815 - 0.1144181141i\)
\(L(1)\) \(\approx\) \(0.8934186815 - 0.1144181141i\)

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

−18.52425120720785877487982960966, −17.92878077933735000029241675915, −17.61409619652940600072930145881, −16.692927444175116713422635125614, −15.82355103747446996037486325918, −15.13043904053597714844044441542, −14.49036668392322618344509377067, −13.98747980235437412825215642768, −13.20102006952882830389954380224, −12.62136930294411365052742161440, −11.33625889339173533351473149307, −11.002845341510845276892075490568, −10.3636229872866212029492417204, −9.17062284692105555784395384218, −8.816863848858264024559929842713, −8.09349327024223486831190822275, −7.347357338193878549936736682885, −6.77955743432934970950050175361, −6.01968104009743001087835377406, −5.46163398754331265332517621311, −4.764317163917827547264319857, −2.95360995657489228179104896565, −2.51008434064876459935057525028, −1.72689189376414025895929794880, −0.914298101705666815544014068994, 0.521640648379566166569962896712, 1.80147275531526534428034471240, 2.0652102461767685440180496922, 3.343163535414059092283986266418, 4.03986426208423703280427460335, 4.603061320497087498012648970220, 5.58042652122358178733193010585, 6.25597219720801382610888558466, 7.71979936900951679850234528940, 8.00866466877964891957593416104, 8.94747585853635649981459752888, 9.23439435593672130517928694796, 10.298805895134891070318480398979, 10.59689897990267078033658629730, 11.114108227802991929794318479118, 12.02095213366178699175586388891, 12.99488943118611965802278119781, 13.48312398398795348141988598131, 14.12712465271107430595201323621, 15.23249441651444756736689338316, 15.709690941708053804570512387317, 16.75335448457616206835657537348, 16.937579231323916807450053947483, 17.53147061386790951505521890375, 18.34598963565275667851114197333

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