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Properties

Label 1-4033-4033.83-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.354 - 0.935i$
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 + (−0.0581 + 0.998i)3-s + (0.173 − 0.984i)4-s + (−0.993 + 0.116i)5-s + (−0.597 − 0.802i)6-s + (0.597 + 0.802i)7-s + (0.5 + 0.866i)8-s + (−0.993 − 0.116i)9-s + (0.686 − 0.727i)10-s + (0.686 + 0.727i)11-s + (0.973 + 0.230i)12-s + (−0.396 + 0.918i)13-s + (−0.973 − 0.230i)14-s + (−0.0581 − 0.998i)15-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (−0.0581 + 0.998i)3-s + (0.173 − 0.984i)4-s + (−0.993 + 0.116i)5-s + (−0.597 − 0.802i)6-s + (0.597 + 0.802i)7-s + (0.5 + 0.866i)8-s + (−0.993 − 0.116i)9-s + (0.686 − 0.727i)10-s + (0.686 + 0.727i)11-s + (0.973 + 0.230i)12-s + (−0.396 + 0.918i)13-s + (−0.973 − 0.230i)14-s + (−0.0581 − 0.998i)15-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.354 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.354 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.354 - 0.935i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.354 - 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4796493014 + 0.6946214475i\)
\(L(\frac12)\) \(\approx\) \(-0.4796493014 + 0.6946214475i\)
\(L(1)\) \(\approx\) \(0.3547793531 + 0.5453424550i\)
\(L(1)\) \(\approx\) \(0.3547793531 + 0.5453424550i\)

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 \( 1 + (-0.0581 + 0.998i)T \)
5 \( 1 + (-0.993 + 0.116i)T \)
7 \( 1 + (0.597 + 0.802i)T \)
11 \( 1 + (0.686 + 0.727i)T \)
13 \( 1 + (-0.396 + 0.918i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (-0.686 + 0.727i)T \)
31 \( 1 + (0.893 + 0.448i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.396 - 0.918i)T \)
53 \( 1 + (-0.973 - 0.230i)T \)
59 \( 1 + (0.0581 + 0.998i)T \)
61 \( 1 + (-0.686 + 0.727i)T \)
67 \( 1 + (0.286 - 0.957i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (-0.286 + 0.957i)T \)
79 \( 1 + (0.686 + 0.727i)T \)
83 \( 1 + (-0.993 + 0.116i)T \)
89 \( 1 + (0.893 - 0.448i)T \)
97 \( 1 + (-0.0581 + 0.998i)T \)
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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

−18.05229768551324608785465633111, −17.28994427309040308375546130866, −17.093993086727887662686553616256, −16.16564533157831086881077238392, −15.39937692732499352490408045072, −14.499732716883558722990680023940, −13.59483841340906548825474502011, −13.1467495999463136992856035088, −12.34566933546978478607966008616, −11.63210161365833652671715672561, −11.152211937794308095861401343453, −10.8264719105332092260959596255, −9.54866015736612809542397097197, −8.85025249443627354762474291417, −8.10404611941402362096887862503, −7.67938586811865491072743238631, −7.05472440825169320786985943394, −6.40180801670946472586147223607, −5.0297996433141115840061482377, −4.342930362806791038260412396145, −3.29820309901415777411840891535, −2.82424716430827853144301964872, −1.74213395187667846237675063355, −0.68639510323701388681601642507, −0.5127009199845487609624657957, 1.27986195719532194919395020775, 2.148268416186750310919245503964, 3.23659843431278509959948304851, 4.26891696069891106004710427963, 4.73289926643179992961406187441, 5.46792378071250835032550154671, 6.432913258583812186998568684041, 7.07221090850367618654396110375, 7.97009923436670486606711400216, 8.59377981886356604758853536531, 9.16443602340253010721746629538, 9.7402703160701756051179362734, 10.63725017891375417662974252788, 11.35653104209704165375850865429, 11.704230274906165701571914546876, 12.52100048617892077434410373962, 13.95497733897406043727647119002, 14.76228545281036564944617792342, 14.87041469739957087888729753497, 15.54902289812590937326073917303, 16.20816096410837523986196117683, 16.8406580128838303032961879522, 17.40797290295040158751906150238, 18.20918083543575440374914448976, 18.86591044891119286615910370550

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