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Properties

Label 1-4033-4033.1877-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.0459 - 0.998i$
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  + (0.642 − 0.766i)2-s + (−0.286 − 0.957i)3-s + (−0.173 − 0.984i)4-s + (0.0581 − 0.998i)5-s + (−0.918 − 0.396i)6-s + (0.893 − 0.448i)7-s + (−0.866 − 0.5i)8-s + (−0.835 + 0.549i)9-s + (−0.727 − 0.686i)10-s + (−0.727 + 0.686i)11-s + (−0.893 + 0.448i)12-s + (0.448 + 0.893i)13-s + (0.230 − 0.973i)14-s + (−0.973 + 0.230i)15-s + (−0.939 + 0.342i)16-s + (0.342 + 0.939i)17-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)2-s + (−0.286 − 0.957i)3-s + (−0.173 − 0.984i)4-s + (0.0581 − 0.998i)5-s + (−0.918 − 0.396i)6-s + (0.893 − 0.448i)7-s + (−0.866 − 0.5i)8-s + (−0.835 + 0.549i)9-s + (−0.727 − 0.686i)10-s + (−0.727 + 0.686i)11-s + (−0.893 + 0.448i)12-s + (0.448 + 0.893i)13-s + (0.230 − 0.973i)14-s + (−0.973 + 0.230i)15-s + (−0.939 + 0.342i)16-s + (0.342 + 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0459 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0459 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.0459 - 0.998i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1877, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.0459 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.159592169 - 1.107498919i\)
\(L(\frac12)\) \(\approx\) \(1.159592169 - 1.107498919i\)
\(L(1)\) \(\approx\) \(0.7199237374 - 0.9237909146i\)
\(L(1)\) \(\approx\) \(0.7199237374 - 0.9237909146i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.642 - 0.766i)T \)
3 \( 1 + (-0.286 - 0.957i)T \)
5 \( 1 + (0.0581 - 0.998i)T \)
7 \( 1 + (0.893 - 0.448i)T \)
11 \( 1 + (-0.727 + 0.686i)T \)
13 \( 1 + (0.448 + 0.893i)T \)
17 \( 1 + (0.342 + 0.939i)T \)
19 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (-0.642 - 0.766i)T \)
29 \( 1 + (-0.286 - 0.957i)T \)
31 \( 1 + (-0.835 + 0.549i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.802 + 0.597i)T \)
53 \( 1 + (0.549 - 0.835i)T \)
59 \( 1 + (0.230 + 0.973i)T \)
61 \( 1 + (-0.993 + 0.116i)T \)
67 \( 1 + (-0.998 + 0.0581i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (-0.686 - 0.727i)T \)
79 \( 1 + (-0.448 + 0.893i)T \)
83 \( 1 + (0.286 + 0.957i)T \)
89 \( 1 + (0.993 - 0.116i)T \)
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

−18.35041974727670783968957311242, −17.69523424303117252931527494253, −17.00406649386287910898757990069, −16.12271671829853611252506668683, −15.73965433188935618731022065112, −14.99628192791711013155771760032, −14.61775644860833337375473513589, −13.91651469648889159815839462802, −13.298756806661101148748886999, −12.0757186878230066344291106532, −11.71224716437267759794090062694, −10.76297729156943173009648254904, −10.49509924941381250676725347223, −9.351275980540921527253419858627, −8.60591743809846116742403983650, −7.858501477125048545303820098715, −7.33919164442412492901556311319, −6.15873009821695486611512454676, −5.531527414548430061857957839600, −5.36888208846002148194824768220, −4.2423872063189847734405246663, −3.42490333734496968129844020477, −3.03432052168501728829268154111, −2.00451077306594659394981268193, −0.26064806895513814363934782340, 0.62063837359931519024860501172, 1.57436926910862909448327737236, 1.82327376062124346306348068315, 2.74446405742875379077520425746, 4.13280335290090373855163819707, 4.496842601748676606144029479631, 5.28590666470262703581881482689, 5.95444711893209960285508404766, 6.76652387919103977361600026739, 7.67253015893086512480679326890, 8.40600615511934659898481716427, 9.02941851944090187648645909010, 10.087973139343073527042183915496, 10.76140660919162206548221210476, 11.48094525958336845551293860789, 12.02488191828154724605484243333, 12.68697798724504903195162081797, 13.28681637165432886294642204166, 13.68131530059070116967604167546, 14.555579258308399885332899113490, 15.1367772963380061370007823802, 16.223013828132508697104702424955, 16.86036639625306886925567322970, 17.7016203538991678826009512433, 18.10856791041585035262789350291

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