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Properties

Label 1-4033-4033.1494-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.890 - 0.455i$
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  + (0.5 − 0.866i)2-s + (0.939 + 0.342i)3-s + (−0.5 − 0.866i)4-s + (−0.642 − 0.766i)5-s + (0.766 − 0.642i)6-s + (0.766 − 0.642i)7-s − 8-s + (0.766 + 0.642i)9-s + (−0.984 + 0.173i)10-s + (0.984 + 0.173i)11-s + (−0.173 − 0.984i)12-s + (−0.766 + 0.642i)13-s + (−0.173 − 0.984i)14-s + (−0.342 − 0.939i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.939 + 0.342i)3-s + (−0.5 − 0.866i)4-s + (−0.642 − 0.766i)5-s + (0.766 − 0.642i)6-s + (0.766 − 0.642i)7-s − 8-s + (0.766 + 0.642i)9-s + (−0.984 + 0.173i)10-s + (0.984 + 0.173i)11-s + (−0.173 − 0.984i)12-s + (−0.766 + 0.642i)13-s + (−0.173 − 0.984i)14-s + (−0.342 − 0.939i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.890 - 0.455i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1494, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.890 - 0.455i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.653538546 - 0.6394546700i\)
\(L(\frac12)\) \(\approx\) \(2.653538546 - 0.6394546700i\)
\(L(1)\) \(\approx\) \(1.535492259 - 0.6377538960i\)
\(L(1)\) \(\approx\) \(1.535492259 - 0.6377538960i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.939 + 0.342i)T \)
5 \( 1 + (-0.642 - 0.766i)T \)
7 \( 1 + (0.766 - 0.642i)T \)
11 \( 1 + (0.984 + 0.173i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.984 - 0.173i)T \)
31 \( 1 + (0.342 + 0.939i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (-0.642 + 0.766i)T \)
53 \( 1 + (-0.984 + 0.173i)T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (0.984 - 0.173i)T \)
67 \( 1 + (0.984 + 0.173i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.173 + 0.984i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (-0.766 + 0.642i)T \)
89 \( 1 + (0.342 - 0.939i)T \)
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

−18.43756959658884728199408608846, −17.82375238842165229029913096959, −17.30490692407768329346262993574, −16.15678222631297728665587217801, −15.59270257019549664469748707537, −14.903531959969776797736020739233, −14.50052906541581751428592303883, −14.17040366828064932332525495674, −13.20917387034380422902267384020, −12.37809337227833165292505056545, −11.94978618512634915333652576517, −11.17613039270831723664365871289, −10.06594523200200555722581621398, −9.16015629665427659539053340203, −8.53216230733772224027296421528, −7.97009323415631909137922614109, −7.39197030463374634427313696705, −6.59168271078140386510729673447, −6.14549226700827387537541471950, −4.855233141093185832082120934311, −4.30813149631488318744754301988, −3.565214522379444254435011128448, −2.59221165398619758600252622608, −2.28630955325411668890462063438, −0.59461195150858834497288490816, 1.09721213516161293407968671959, 1.69156711016238051377739638297, 2.441628469789687545588697653176, 3.60276236126109851118894090708, 4.23996414287272205994929954223, 4.39715665017877291471524056835, 5.23126205533553175206223585696, 6.4932276858230025400165286831, 7.30389261585460274579392583993, 8.39254490948301011040937234536, 8.5726940633997568793256306805, 9.576693396379053645234360421537, 10.0101323398672730139787969855, 11.069362160194061583398552947931, 11.426623672402651699810845558722, 12.49003507311231623407915407473, 12.73241599656566240656724412785, 13.80196993983135068300087235189, 14.25716577026496490597392753739, 14.76059011051581225666663286512, 15.49751643512647439722146125068, 16.19293548956003068110219106108, 17.33730108947365279261980668479, 17.50643764491949095941407701935, 18.94219610886776698912674519426

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