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Properties

Label 1-4033-4033.71-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.928 + 0.371i$
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 + (−0.5 + 0.866i)3-s + (0.173 − 0.984i)4-s + 5-s + (−0.173 − 0.984i)6-s + 7-s + (0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + (−0.766 + 0.642i)10-s + (−0.766 − 0.642i)11-s + (0.766 + 0.642i)12-s + (0.5 − 0.866i)13-s + (−0.766 + 0.642i)14-s + (−0.5 + 0.866i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (−0.5 + 0.866i)3-s + (0.173 − 0.984i)4-s + 5-s + (−0.173 − 0.984i)6-s + 7-s + (0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + (−0.766 + 0.642i)10-s + (−0.766 − 0.642i)11-s + (0.766 + 0.642i)12-s + (0.5 − 0.866i)13-s + (−0.766 + 0.642i)14-s + (−0.5 + 0.866i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.928 + 0.371i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.928 + 0.371i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.413994798 + 0.2720627728i\)
\(L(\frac12)\) \(\approx\) \(1.413994798 + 0.2720627728i\)
\(L(1)\) \(\approx\) \(0.8507847704 + 0.2763161742i\)
\(L(1)\) \(\approx\) \(0.8507847704 + 0.2763161742i\)

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 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 + (-0.766 - 0.642i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (0.766 + 0.642i)T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.173 + 0.984i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (0.766 + 0.642i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (0.173 + 0.984i)T \)
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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

−18.37529038769787813292870694676, −17.7275940242977479454838245351, −17.48597190026054227707803648510, −16.68928258951703825330124986044, −16.11313251112773169781327880438, −14.90654400960593649644771904779, −14.152878348264777386221440507473, −13.276822624225307381527037023092, −13.042509494036500320519636017069, −12.08106840819294880005226072540, −11.53852739673502538212734841399, −10.847412478025918604492298424284, −10.26373826246676954664367916809, −9.52865219409082385560985721477, −8.590172993646910137274200457599, −8.049632234505453978159159298300, −7.39034503644233773326860064296, −6.52396726399116298355975104990, −5.9241135073007723382965474647, −4.88954109544606588968074483586, −4.27737687801554306769892374502, −2.80980610259710487388617559650, −2.21167986858405934483530590656, −1.56880952939914321244051900332, −0.98983108945255060952424588492, 0.68458275199927624319163811193, 1.36712857844632676391057219308, 2.58498969800115733178689308911, 3.33545882534529050970913497927, 4.89174325011209070746585033270, 5.17074904419277317488501469242, 5.61982685343410868087689416920, 6.495616893361175033738333823255, 7.34168857971596014505201619464, 8.19616382383091690921172181114, 8.88871639466094449591367995450, 9.451556230167378020047960257160, 10.21026139116421305917986528022, 10.84281467286183403157251280568, 11.162555655861087494095322741129, 12.106198757638464289024823877293, 13.413768315076343320827424765929, 13.8840878588810889131475616880, 14.56080571406570362253631251621, 15.38686338302585974893843771733, 15.87333158346262924888690243215, 16.44057034047722047282804020588, 17.34373303705577818620982202659, 17.71516277777042155077912465385, 18.16422570602582969811376325155

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