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Properties

Label 1-4033-4033.659-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.768 + 0.639i$
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  − 2-s + (−0.286 − 0.957i)3-s + 4-s + (−0.893 + 0.448i)5-s + (0.286 + 0.957i)6-s + (−0.835 − 0.549i)7-s − 8-s + (−0.835 + 0.549i)9-s + (0.893 − 0.448i)10-s + (0.893 + 0.448i)11-s + (−0.286 − 0.957i)12-s + (−0.893 + 0.448i)13-s + (0.835 + 0.549i)14-s + (0.686 + 0.727i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (−0.286 − 0.957i)3-s + 4-s + (−0.893 + 0.448i)5-s + (0.286 + 0.957i)6-s + (−0.835 − 0.549i)7-s − 8-s + (−0.835 + 0.549i)9-s + (0.893 − 0.448i)10-s + (0.893 + 0.448i)11-s + (−0.286 − 0.957i)12-s + (−0.893 + 0.448i)13-s + (0.835 + 0.549i)14-s + (0.686 + 0.727i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.768 + 0.639i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.768 + 0.639i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09644636706 + 0.2665773927i\)
\(L(\frac12)\) \(\approx\) \(0.09644636706 + 0.2665773927i\)
\(L(1)\) \(\approx\) \(0.4467266292 + 0.002979681825i\)
\(L(1)\) \(\approx\) \(0.4467266292 + 0.002979681825i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.286 - 0.957i)T \)
5 \( 1 + (-0.893 + 0.448i)T \)
7 \( 1 + (-0.835 - 0.549i)T \)
11 \( 1 + (0.893 + 0.448i)T \)
13 \( 1 + (-0.893 + 0.448i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.835 + 0.549i)T \)
31 \( 1 + (-0.597 + 0.802i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (-0.286 + 0.957i)T \)
53 \( 1 + (0.973 + 0.230i)T \)
59 \( 1 + (0.686 + 0.727i)T \)
61 \( 1 + (-0.396 + 0.918i)T \)
67 \( 1 + (-0.286 + 0.957i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.835 + 0.549i)T \)
79 \( 1 + (0.686 + 0.727i)T \)
83 \( 1 + (-0.993 + 0.116i)T \)
89 \( 1 + (0.0581 - 0.998i)T \)
97 \( 1 + (-0.597 - 0.802i)T \)
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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

−18.25652060601219207477277371115, −17.1499733338129361167231893391, −16.86765067280402828417037845288, −16.299984754898887525555589494641, −15.64539651344003072661683393350, −15.05840757322332489762726505431, −14.65609948651664270946749977330, −13.25814244023018827377689709972, −12.286852928548580924298605351929, −11.872195741694552489480758157754, −11.313148221360595695944801526590, −10.50848868418917414423645132679, −9.73061318204553702754374995018, −9.224742214348037833854571521702, −8.678735085097918311905611325364, −7.98281763653367938207575421572, −6.917048023004169137001014973962, −6.44656115913382020761814358198, −5.40606084484067637998753613562, −4.796358694414110973621461447363, −3.66349056302946103491205521322, −3.14599334670011490780151688278, −2.32985992585366163693481999276, −0.75387873874217239177513937887, −0.18675507166155292593169061918, 1.05001298567243006780838112654, 1.71806612632076845301702163572, 2.78653803529839099973784527871, 3.438124864532067478954153816657, 4.37904363286669003161447598857, 5.673055603003484381658866580603, 6.537187222663164858029437062798, 7.09981383973392408400158823168, 7.22642998319596095691998931762, 8.32796426532716930210213664505, 8.78004598279106035841177476518, 9.884269563984183441505310008822, 10.41986991435542298122823786918, 11.145000018091831071293597477465, 11.94375355217782222907713735967, 12.33733697321960092308077196565, 12.92759303922705453497908082469, 14.17605499153276618520722670803, 14.709444093169794000327811561701, 15.36035549102670087281302002754, 16.53334535936826152948924813352, 16.689843943636213701613702683991, 17.34603958757282596462694390575, 18.11327773289859509432376356174, 18.94562666136994491481315928775

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