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Properties

Label 1-4033-4033.3560-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.723 - 0.690i$
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 + (−0.893 − 0.448i)3-s + (−0.939 + 0.342i)4-s + (−0.918 + 0.396i)5-s + (0.286 − 0.957i)6-s + (−0.993 − 0.116i)7-s + (−0.5 − 0.866i)8-s + (0.597 + 0.802i)9-s + (−0.549 − 0.835i)10-s + (0.549 − 0.835i)11-s + (0.993 + 0.116i)12-s + (−0.993 − 0.116i)13-s + (−0.0581 − 0.998i)14-s + (0.998 + 0.0581i)15-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.893 − 0.448i)3-s + (−0.939 + 0.342i)4-s + (−0.918 + 0.396i)5-s + (0.286 − 0.957i)6-s + (−0.993 − 0.116i)7-s + (−0.5 − 0.866i)8-s + (0.597 + 0.802i)9-s + (−0.549 − 0.835i)10-s + (0.549 − 0.835i)11-s + (0.993 + 0.116i)12-s + (−0.993 − 0.116i)13-s + (−0.0581 − 0.998i)14-s + (0.998 + 0.0581i)15-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.723 - 0.690i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3560, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.723 - 0.690i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1334641032 - 0.05346978801i\)
\(L(\frac12)\) \(\approx\) \(0.1334641032 - 0.05346978801i\)
\(L(1)\) \(\approx\) \(0.4386346795 + 0.2145865163i\)
\(L(1)\) \(\approx\) \(0.4386346795 + 0.2145865163i\)

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 \( 1 + (-0.893 - 0.448i)T \)
5 \( 1 + (-0.918 + 0.396i)T \)
7 \( 1 + (-0.993 - 0.116i)T \)
11 \( 1 + (0.549 - 0.835i)T \)
13 \( 1 + (-0.993 - 0.116i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (-0.448 + 0.893i)T \)
31 \( 1 + (-0.802 + 0.597i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.642 - 0.766i)T \)
47 \( 1 + (-0.230 + 0.973i)T \)
53 \( 1 + (0.802 + 0.597i)T \)
59 \( 1 + (-0.0581 + 0.998i)T \)
61 \( 1 + (0.727 - 0.686i)T \)
67 \( 1 + (0.918 + 0.396i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (0.835 + 0.549i)T \)
79 \( 1 + (-0.993 + 0.116i)T \)
83 \( 1 + (-0.893 - 0.448i)T \)
89 \( 1 + (-0.727 + 0.686i)T \)
97 \( 1 + (-0.918 + 0.396i)T \)
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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

−18.516046027887073590974092121581, −18.107478810645039590887212204600, −17.05664770063556563820904358052, −16.63743521654331532496112975373, −15.9217052790314526794009754624, −15.0436799434523848695136964331, −14.68818198286602304297028793170, −13.45001383046751474305245593376, −12.69700181729676481216809784856, −12.31470315983623163476552435671, −11.65719405613708457066114619229, −11.28392901615475784598245234939, −10.199931725139138291974484509496, −9.64532039568932533598336072143, −9.33093814176946143900247697059, −8.268809469940883598157527782068, −7.21166648479042160673420084801, −6.57180846363287164759247395853, −5.55645589830254225274418413832, −4.84873054469164081526674576605, −4.23300992509425135189776284122, −3.70111699138600714970594839090, −2.7745308087558848487064803288, −1.76532821019638998972653525077, −0.563354571493904004749197397363, 0.08971058190701705202235862500, 1.12539009677190894632184208230, 2.69184812555885932136631300991, 3.61855341973096173473120954028, 4.12167623976143304664679449503, 5.22315985408235343515730111519, 5.68804799456803044883838266261, 6.68704833565419420240013380569, 7.00802023984149471337786881497, 7.52022411625930292402325118833, 8.49561164047781390510761768255, 9.224318591344852094883206674762, 10.08834313766901991101892460106, 10.92544616881408749720943699928, 11.6360536354535862516413911142, 12.372291384952672367770535477630, 12.87425855686411907343730771934, 13.64349934813746634206533676311, 14.307313218035790418710316260297, 15.23087453531987855304951203982, 15.82854346872685924080967392120, 16.245492454706959111671097938100, 17.01415886709461256539779352100, 17.463168626613246286190086186866, 18.34838009266806866266674843338

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