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Properties

Label 1-4033-4033.2694-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.762 + 0.647i$
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  + (−0.173 + 0.984i)2-s + (0.597 − 0.802i)3-s + (−0.939 − 0.342i)4-s + (0.286 − 0.957i)5-s + (0.686 + 0.727i)6-s + (−0.686 − 0.727i)7-s + (0.5 − 0.866i)8-s + (−0.286 − 0.957i)9-s + (0.893 + 0.448i)10-s + (0.893 − 0.448i)11-s + (−0.835 + 0.549i)12-s + (−0.973 + 0.230i)13-s + (0.835 − 0.549i)14-s + (−0.597 − 0.802i)15-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.597 − 0.802i)3-s + (−0.939 − 0.342i)4-s + (0.286 − 0.957i)5-s + (0.686 + 0.727i)6-s + (−0.686 − 0.727i)7-s + (0.5 − 0.866i)8-s + (−0.286 − 0.957i)9-s + (0.893 + 0.448i)10-s + (0.893 − 0.448i)11-s + (−0.835 + 0.549i)12-s + (−0.973 + 0.230i)13-s + (0.835 − 0.549i)14-s + (−0.597 − 0.802i)15-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.762 + 0.647i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2694, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.762 + 0.647i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05985081847 - 0.1629604230i\)
\(L(\frac12)\) \(\approx\) \(-0.05985081847 - 0.1629604230i\)
\(L(1)\) \(\approx\) \(0.8291044153 - 0.1492856775i\)
\(L(1)\) \(\approx\) \(0.8291044153 - 0.1492856775i\)

Euler product

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

−18.903370004112170783516983058241, −18.48943128750599349134716251617, −17.69026040596365228956958280066, −16.90064613472371499521962406404, −16.22945996062307834057044611183, −15.191028665796742088698122363723, −14.825128305168816530381689997159, −14.00472210674735133436605306744, −13.626881593821659408398302394490, −12.55346554311281274125917211565, −11.90720500449882139830038194288, −11.322452387904070697794357955, −10.39551319318576223316493444544, −9.870174438522356754481375178286, −9.35834559484131438695624609411, −9.001603018434822669636991858920, −7.69922686078441146971394485818, −7.32151103740840122498174818299, −5.98525364269169192280805012244, −5.35020532295019279564126899671, −4.400569111641117083068476608313, −3.52706201518195855276060741549, −3.16683791396756708992865752047, −2.245762785369756661391955361220, −1.85940116210224174399197406396, 0.04954617235711873307586027656, 1.02271704195756780233041678220, 1.69442742496450416884496399779, 2.94337524933793578890190742694, 3.98957398699140682680867847283, 4.42905557053883757242568381886, 5.58093373055086911986366547727, 6.26124406329070081614845861527, 6.82835518809706558885231600836, 7.53880672880858588822154163046, 8.2340658080299799313527063922, 8.92735711359577529651255410579, 9.478171096940225442767814120077, 9.98175242101144557266143789037, 11.21241691039167263254017090441, 12.33622513818933761634135521808, 12.76606221849667479489570191403, 13.35021191237130744874367124856, 14.14840088868126601917318928599, 14.35825021959469086319899532222, 15.39647383344609932412375620987, 16.09987409483232096694243549865, 16.97542439201672631422328031354, 17.10002901896130224734939975705, 17.84919472594815013814988646622

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