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Properties

Label 1-4033-4033.1039-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.961 + 0.274i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  − i·2-s + (−0.835 − 0.549i)3-s − 4-s + (−0.597 − 0.802i)5-s + (−0.549 + 0.835i)6-s + (0.396 − 0.918i)7-s + i·8-s + (0.396 + 0.918i)9-s + (−0.802 + 0.597i)10-s + (−0.802 − 0.597i)11-s + (0.835 + 0.549i)12-s + (−0.802 + 0.597i)13-s + (−0.918 − 0.396i)14-s + (0.0581 + 0.998i)15-s + 16-s + (−0.866 + 0.5i)17-s + ⋯
L(s)  = 1  − i·2-s + (−0.835 − 0.549i)3-s − 4-s + (−0.597 − 0.802i)5-s + (−0.549 + 0.835i)6-s + (0.396 − 0.918i)7-s + i·8-s + (0.396 + 0.918i)9-s + (−0.802 + 0.597i)10-s + (−0.802 − 0.597i)11-s + (0.835 + 0.549i)12-s + (−0.802 + 0.597i)13-s + (−0.918 − 0.396i)14-s + (0.0581 + 0.998i)15-s + 16-s + (−0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.961 + 0.274i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.961 + 0.274i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04192466489 - 0.2999244426i\)
\(L(\frac12)\) \(\approx\) \(-0.04192466489 - 0.2999244426i\)
\(L(1)\) \(\approx\) \(0.3269443769 - 0.3727720030i\)
\(L(1)\) \(\approx\) \(0.3269443769 - 0.3727720030i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - iT \)
3 \( 1 + (-0.835 - 0.549i)T \)
5 \( 1 + (-0.597 - 0.802i)T \)
7 \( 1 + (0.396 - 0.918i)T \)
11 \( 1 + (-0.802 - 0.597i)T \)
13 \( 1 + (-0.802 + 0.597i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.984 + 0.173i)T \)
23 \( 1 + (-0.642 - 0.766i)T \)
29 \( 1 + (0.396 - 0.918i)T \)
31 \( 1 + (-0.286 + 0.957i)T \)
41 \( 1 + (-0.642 + 0.766i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (-0.549 - 0.835i)T \)
53 \( 1 + (0.448 + 0.893i)T \)
59 \( 1 + (0.998 - 0.0581i)T \)
61 \( 1 + (-0.686 + 0.727i)T \)
67 \( 1 + (0.549 + 0.835i)T \)
71 \( 1 - T \)
73 \( 1 + (0.396 + 0.918i)T \)
79 \( 1 + (0.998 - 0.0581i)T \)
83 \( 1 + (-0.973 - 0.230i)T \)
89 \( 1 + (0.993 - 0.116i)T \)
97 \( 1 + (0.286 + 0.957i)T \)
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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

−18.21027997913322898029679551730, −17.94754805596700823647227331014, −17.43967765722616447521807208167, −16.49563395746331788070405661215, −15.6750740337483710774949623910, −15.44540672552937739031242624352, −14.89515482608979073384148816848, −14.328226219706674641490072128734, −13.10965587820141344004767457286, −12.52837859360983825718657557223, −11.81335157218117984231313037630, −11.05454455276695398131733844246, −10.334211524842827251186821299741, −9.688033744147460991497809040953, −8.91574548959811499131383151491, −7.99942184309937269742101542173, −7.446143669347510007133101525607, −6.640199745459509346495619090856, −6.048294291706172310065586135200, −5.12192346091789827471384230261, −4.80697730110024351917717611837, −3.94445338023355499121676073322, −2.98394121102733479078048799508, −2.02315154227895553235208273371, −0.3259342446858542764088851622, 0.17125747732472466308203082144, 0.87363167961428263645324321832, 1.79388769128997425210067132026, 2.430746231205350290420920575708, 3.78615775615260982498051161025, 4.43776198424352628383232138998, 4.841519119398397262858920868364, 5.700955254310994467099187313548, 6.70641024285030378012723601781, 7.55653665708399979588652223026, 8.33238019419060717713614029981, 8.67674362559616969433343412806, 10.02578327794246487690228817688, 10.4773898527856699831868497452, 11.13224781514018381200740133447, 11.77568920574676609376496717026, 12.32180037123909779840857639522, 13.05994016740722454457396042923, 13.44829429773941071895885628636, 14.22811326175726724796419355213, 15.182822464195629725557727518152, 16.23501516371155129513843751563, 16.77433061137020422024982494344, 17.31049491547737103561659998756, 17.923858149544148833261783788407

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