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Properties

Label 1-4033-4033.208-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.983 - 0.183i$
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 + (0.686 − 0.727i)3-s + (0.173 + 0.984i)4-s + (0.448 + 0.893i)5-s + (0.993 − 0.116i)6-s + (−0.835 − 0.549i)7-s + (−0.5 + 0.866i)8-s + (−0.0581 − 0.998i)9-s + (−0.230 + 0.973i)10-s + (0.230 + 0.973i)11-s + (0.835 + 0.549i)12-s + (−0.835 − 0.549i)13-s + (−0.286 − 0.957i)14-s + (0.957 + 0.286i)15-s + (−0.939 + 0.342i)16-s + (0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.686 − 0.727i)3-s + (0.173 + 0.984i)4-s + (0.448 + 0.893i)5-s + (0.993 − 0.116i)6-s + (−0.835 − 0.549i)7-s + (−0.5 + 0.866i)8-s + (−0.0581 − 0.998i)9-s + (−0.230 + 0.973i)10-s + (0.230 + 0.973i)11-s + (0.835 + 0.549i)12-s + (−0.835 − 0.549i)13-s + (−0.286 − 0.957i)14-s + (0.957 + 0.286i)15-s + (−0.939 + 0.342i)16-s + (0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.983 - 0.183i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.983 - 0.183i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1050153949 + 1.136454258i\)
\(L(\frac12)\) \(\approx\) \(-0.1050153949 + 1.136454258i\)
\(L(1)\) \(\approx\) \(1.363234765 + 0.5982206425i\)
\(L(1)\) \(\approx\) \(1.363234765 + 0.5982206425i\)

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 \( 1 + (0.686 - 0.727i)T \)
5 \( 1 + (0.448 + 0.893i)T \)
7 \( 1 + (-0.835 - 0.549i)T \)
11 \( 1 + (0.230 + 0.973i)T \)
13 \( 1 + (-0.835 - 0.549i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (-0.727 - 0.686i)T \)
31 \( 1 + (0.998 - 0.0581i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (-0.918 + 0.396i)T \)
53 \( 1 + (-0.998 - 0.0581i)T \)
59 \( 1 + (-0.286 + 0.957i)T \)
61 \( 1 + (-0.802 + 0.597i)T \)
67 \( 1 + (-0.448 + 0.893i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (-0.973 + 0.230i)T \)
79 \( 1 + (-0.835 + 0.549i)T \)
83 \( 1 + (0.686 - 0.727i)T \)
89 \( 1 + (0.802 - 0.597i)T \)
97 \( 1 + (0.448 + 0.893i)T \)
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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

−18.51402743636208922770928746019, −17.14357974665698765187536384733, −16.48394349366710948361467514443, −16.0938936730169992678828250140, −15.23877611195155479604223195919, −14.56469074018172371836394763551, −13.994258022757574812217607771831, −13.33970597980969068157288065622, −12.65865667015552638960056997982, −12.14229118273569074528819551234, −11.29309725857914454051427989240, −10.31255681841372833035040295034, −9.91407558885028082856831294567, −9.14093877312641563597643944507, −8.76336096740323487607567023798, −7.811282800456509503941069242701, −6.41330088893664838276277566917, −6.003702285421674202692765478448, −5.06309572194482534113169711978, −4.58479937068956415550380103416, −3.68553129105300161120802094480, −3.07519593012189824380309447355, −2.26831499343034282937464426250, −1.584733053680445646473078628400, −0.18237055008748813605860655490, 1.57147557057651001188238465422, 2.48968215909745561861482257002, 2.9887500227846437701375585969, 3.75956501083250077278787860330, 4.48347829736794929047382536043, 5.77675967526791953925836514324, 6.20395700110513723828298258601, 6.96872447713999775096601536125, 7.545595755365410897834059755371, 7.85568272445303243682392248408, 9.12393812071107955053178881854, 9.83414438544980250883880145874, 10.33974720041239430916613978455, 11.67213531244641301437932056134, 12.19914791241965852914601396439, 12.988324052171945843493157544704, 13.33240348231874602626118268701, 14.27686069676456424995915456357, 14.51428431859637732686664545546, 15.205492181159318612863279399610, 15.860580696284167271701808360540, 16.943713649330458016232642816904, 17.44622042075042985690827135258, 17.92842505920951143359249325772, 18.95965433896051527274163516299

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