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Properties

Label 1-4033-4033.405-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.711 - 0.703i$
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  + (−0.866 + 0.5i)2-s + (−0.973 + 0.230i)3-s + (0.5 − 0.866i)4-s + (0.998 − 0.0581i)5-s + (0.727 − 0.686i)6-s + (0.893 + 0.448i)7-s + i·8-s + (0.893 − 0.448i)9-s + (−0.835 + 0.549i)10-s + (0.835 + 0.549i)11-s + (−0.286 + 0.957i)12-s + (0.998 − 0.0581i)13-s + (−0.998 + 0.0581i)14-s + (−0.957 + 0.286i)15-s + (−0.5 − 0.866i)16-s + i·17-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.973 + 0.230i)3-s + (0.5 − 0.866i)4-s + (0.998 − 0.0581i)5-s + (0.727 − 0.686i)6-s + (0.893 + 0.448i)7-s + i·8-s + (0.893 − 0.448i)9-s + (−0.835 + 0.549i)10-s + (0.835 + 0.549i)11-s + (−0.286 + 0.957i)12-s + (0.998 − 0.0581i)13-s + (−0.998 + 0.0581i)14-s + (−0.957 + 0.286i)15-s + (−0.5 − 0.866i)16-s + i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.711 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.711 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.711 - 0.703i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (405, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.711 - 0.703i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04939394008 + 0.1202235623i\)
\(L(\frac12)\) \(\approx\) \(-0.04939394008 + 0.1202235623i\)
\(L(1)\) \(\approx\) \(0.6684598258 + 0.2221344103i\)
\(L(1)\) \(\approx\) \(0.6684598258 + 0.2221344103i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.973 + 0.230i)T \)
5 \( 1 + (0.998 - 0.0581i)T \)
7 \( 1 + (0.893 + 0.448i)T \)
11 \( 1 + (0.835 + 0.549i)T \)
13 \( 1 + (0.998 - 0.0581i)T \)
17 \( 1 + iT \)
19 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (-0.342 - 0.939i)T \)
29 \( 1 + (-0.998 + 0.0581i)T \)
31 \( 1 + (-0.918 - 0.396i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (0.642 + 0.766i)T \)
47 \( 1 + (-0.686 + 0.727i)T \)
53 \( 1 + (0.973 - 0.230i)T \)
59 \( 1 + (0.957 - 0.286i)T \)
61 \( 1 + (-0.802 - 0.597i)T \)
67 \( 1 + (0.286 + 0.957i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.0581 - 0.998i)T \)
79 \( 1 + (-0.727 - 0.686i)T \)
83 \( 1 + (-0.993 - 0.116i)T \)
89 \( 1 + (-0.998 + 0.0581i)T \)
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

−18.01189754583706346686244678917, −17.23169386573049861462031876382, −16.75608488842694963218282651513, −16.34323772435724212934917911713, −15.371423685368076421332374881480, −14.25345376187058275528476225747, −13.62305442816202823468677245504, −13.058026566786274312028860059175, −12.03457750249202805346676391398, −11.57512158648871032671465783258, −10.90531341589477421817683061750, −10.459572304378327398723247735399, −9.64964147402935254261725234511, −8.95384073528256129472634482879, −8.20498045304558885809012990463, −7.24160273790312283256024167147, −6.808137693521170678480126565, −5.81361041034248736358749910301, −5.39462267725180095955304670533, −4.116053726983001110895834090787, −3.56479340701155538886051882135, −2.190828430007101675794548374852, −1.476700659877846739222945771642, −1.14444006942498639959272449574, −0.02861051595196086469399048014, 1.26136896980024355189074711497, 1.555100386591084132146323234769, 2.43887876772411716482969173062, 4.0181326012201514626044001147, 4.78763243444824354239165847546, 5.54632747244235765386033704982, 6.14020508076255631369178962014, 6.59310717323811780975291871212, 7.45417343594072385531372474328, 8.52509941074424427007355154371, 8.970555210761316717978512361447, 9.739471815581726147413656119106, 10.3884862959081002057290984231, 11.16949054785612016428723539768, 11.40507200767555089864883465069, 12.54151199139292101225174201096, 13.156218795825260425696385898, 14.280110558495095983595661718592, 14.84569926467855339045726841676, 15.329736103830649504606292728553, 16.34240446352998383663720864361, 16.78827810221662858475348372301, 17.505781490181034730516834053137, 17.81890674521406989335613784286, 18.38808567482371437188338254428

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