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Properties

Label 1-4033-4033.1731-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.886 - 0.462i$
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  + (−0.939 + 0.342i)2-s + (−0.893 + 0.448i)3-s + (0.766 − 0.642i)4-s + (−0.918 − 0.396i)5-s + (0.686 − 0.727i)6-s + (−0.993 + 0.116i)7-s + (−0.5 + 0.866i)8-s + (0.597 − 0.802i)9-s + (0.998 + 0.0581i)10-s + (−0.998 + 0.0581i)11-s + (−0.396 + 0.918i)12-s + (−0.993 + 0.116i)13-s + (0.893 − 0.448i)14-s + (0.998 − 0.0581i)15-s + (0.173 − 0.984i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (−0.893 + 0.448i)3-s + (0.766 − 0.642i)4-s + (−0.918 − 0.396i)5-s + (0.686 − 0.727i)6-s + (−0.993 + 0.116i)7-s + (−0.5 + 0.866i)8-s + (0.597 − 0.802i)9-s + (0.998 + 0.0581i)10-s + (−0.998 + 0.0581i)11-s + (−0.396 + 0.918i)12-s + (−0.993 + 0.116i)13-s + (0.893 − 0.448i)14-s + (0.998 − 0.0581i)15-s + (0.173 − 0.984i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.886 - 0.462i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1731, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.886 - 0.462i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02412922814 + 0.09842955384i\)
\(L(\frac12)\) \(\approx\) \(0.02412922814 + 0.09842955384i\)
\(L(1)\) \(\approx\) \(0.2978690757 + 0.09338851560i\)
\(L(1)\) \(\approx\) \(0.2978690757 + 0.09338851560i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.939 + 0.342i)T \)
3 \( 1 + (-0.893 + 0.448i)T \)
5 \( 1 + (-0.918 - 0.396i)T \)
7 \( 1 + (-0.993 + 0.116i)T \)
11 \( 1 + (-0.998 + 0.0581i)T \)
13 \( 1 + (-0.993 + 0.116i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (-0.549 + 0.835i)T \)
31 \( 1 + (0.918 - 0.396i)T \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (0.957 + 0.286i)T \)
53 \( 1 + (0.116 + 0.993i)T \)
59 \( 1 + (-0.0581 - 0.998i)T \)
61 \( 1 + (0.727 + 0.686i)T \)
67 \( 1 + (-0.802 - 0.597i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (0.0581 + 0.998i)T \)
79 \( 1 + (0.396 + 0.918i)T \)
83 \( 1 + (0.0581 - 0.998i)T \)
89 \( 1 + (0.957 - 0.286i)T \)
97 \( 1 + (0.116 + 0.993i)T \)
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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

−18.143201792847522863895109096439, −17.49446288147816146416224527945, −16.76023823826825198128617729447, −16.2497996048907257301743275295, −15.47106635008125392948347431352, −15.23882210089259402072996950480, −13.629988718715551876623516720487, −13.08671234161352221698084346676, −12.2989493311036098968127132274, −11.819591588417442915109304007849, −11.23792090357842301896301941687, −10.32450453742467610636970410510, −10.12392525389546744238999996493, −9.14172247553706452780058556392, −8.10572102154585893958548839159, −7.557449770531638099261432188979, −6.96313670693547487095875299200, −6.47317003266600855511904075121, −5.42350239214744976079088906425, −4.52045019883304059631229636777, −3.52518067826798701540578208012, −2.698319208790992966282899984774, −2.098247971332113991545834311592, −0.60428766280731906132474026326, −0.10693714194621375313300156157, 0.74602578109306762860942713912, 2.039345515749881232673461671534, 2.99900870238511627338918443218, 4.01577630710957812800639744626, 4.79932745608651281965534679151, 5.55189346438642233381066200442, 6.35498628636752398828036957848, 6.91437698444569665199446573101, 7.725268174455908175389639026045, 8.47139131729373315460494843607, 9.154359765052285046181387655025, 10.05330332468105256193318953259, 10.428259855780016324813590544473, 11.0668224315485790188812331120, 12.011156039607417970680495162109, 12.48287106927954769604314079290, 13.03518549622026268498096147039, 14.53767488024075718294834802648, 15.23956387741707368699263458390, 15.58158272597575683670467267257, 16.27603763403901551519314754301, 16.91098504854126613380782908770, 17.12524148603356211178991115264, 18.24143820073054702356621324048, 18.82958295337773596043132617374

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