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Properties

Label 1-4033-4033.20-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.917 - 0.398i$
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  + (0.984 − 0.173i)2-s + (−0.973 − 0.230i)3-s + (0.939 − 0.342i)4-s + (−0.448 − 0.893i)5-s + (−0.998 − 0.0581i)6-s + (−0.835 − 0.549i)7-s + (0.866 − 0.5i)8-s + (0.893 + 0.448i)9-s + (−0.597 − 0.802i)10-s + (0.597 − 0.802i)11-s + (−0.993 + 0.116i)12-s + (−0.998 − 0.0581i)13-s + (−0.918 − 0.396i)14-s + (0.230 + 0.973i)15-s + (0.766 − 0.642i)16-s + (−0.984 + 0.173i)17-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)2-s + (−0.973 − 0.230i)3-s + (0.939 − 0.342i)4-s + (−0.448 − 0.893i)5-s + (−0.998 − 0.0581i)6-s + (−0.835 − 0.549i)7-s + (0.866 − 0.5i)8-s + (0.893 + 0.448i)9-s + (−0.597 − 0.802i)10-s + (0.597 − 0.802i)11-s + (−0.993 + 0.116i)12-s + (−0.998 − 0.0581i)13-s + (−0.918 − 0.396i)14-s + (0.230 + 0.973i)15-s + (0.766 − 0.642i)16-s + (−0.984 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.917 - 0.398i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.917 - 0.398i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9713741483 - 0.2016779941i\)
\(L(\frac12)\) \(\approx\) \(0.9713741483 - 0.2016779941i\)
\(L(1)\) \(\approx\) \(0.9053158020 - 0.4514334450i\)
\(L(1)\) \(\approx\) \(0.9053158020 - 0.4514334450i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.984 - 0.173i)T \)
3 \( 1 + (-0.973 - 0.230i)T \)
5 \( 1 + (-0.448 - 0.893i)T \)
7 \( 1 + (-0.835 - 0.549i)T \)
11 \( 1 + (0.597 - 0.802i)T \)
13 \( 1 + (-0.998 - 0.0581i)T \)
17 \( 1 + (-0.984 + 0.173i)T \)
19 \( 1 + iT \)
23 \( 1 + (-0.984 + 0.173i)T \)
29 \( 1 + (-0.802 + 0.597i)T \)
31 \( 1 + (0.727 - 0.686i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (-0.893 - 0.448i)T \)
53 \( 1 + (0.993 - 0.116i)T \)
59 \( 1 + (0.230 + 0.973i)T \)
61 \( 1 + (-0.116 - 0.993i)T \)
67 \( 1 + (0.597 - 0.802i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (0.993 + 0.116i)T \)
79 \( 1 + (-0.918 + 0.396i)T \)
83 \( 1 + (-0.0581 + 0.998i)T \)
89 \( 1 + (-0.230 + 0.973i)T \)
97 \( 1 + (-0.957 + 0.286i)T \)
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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

−18.17539630313683714010033890791, −17.5600030498005328364621601799, −16.8873316155007785857586986680, −16.06823392178697106292800214154, −15.499440657988134786023713135258, −15.1151607312473354513042577518, −14.450390459651850486952389418878, −13.437148908112248021468671204201, −12.8014676794761007361426065584, −12.01516580836885353886841604179, −11.70187698120707257166395399203, −11.081286289821498787688876443, −10.02385438041196126640232242800, −9.79475994672326334299658879670, −8.50429678197265218954853602560, −7.29645503132022725378293822317, −6.859932100589474599380333418156, −6.46195716677125033766237480476, −5.65594770732385595364379333531, −4.72303722787794931091102554619, −4.27672718536205109432576855939, −3.39858083651976916496590124708, −2.56515407989757721124830452379, −1.86159221545751142279309153331, −0.20150919448999846817732973599, 0.48040909924535649539666375263, 1.40277678349821577737198497851, 2.21077232925828079490897121888, 3.63807249148908594608707145109, 3.89619227455935639843148912358, 4.80713101201985211018940722990, 5.42690791310441865089555652738, 6.2164262972029228954992916685, 6.73468966962734662629304514390, 7.52626048454954583931877633061, 8.30598350381206625972275981836, 9.55663787004483098840784343752, 10.09698088644300714737595173937, 10.926940142114515020418981577656, 11.65485501403465502325958689426, 12.14066765042419766350812538924, 12.67970498174005577578520898063, 13.40704449093699856660200304101, 13.80922048219861263872005921064, 14.93922077233888062546136931594, 15.6217428685885097088004388259, 16.31823477193955762169892598904, 16.82012372958187995712157690929, 17.07072977170337146245887755112, 18.36270293013693434679541975184

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