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Properties

Label 1-4033-4033.22-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.569 + 0.821i$
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  − i·2-s + (0.286 − 0.957i)3-s − 4-s + (−0.448 + 0.893i)5-s + (−0.957 − 0.286i)6-s + (−0.835 + 0.549i)7-s + i·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 + (−0.448 + 0.893i)13-s + (0.549 + 0.835i)14-s + (0.727 + 0.686i)15-s + 16-s + (−0.866 − 0.5i)17-s + ⋯
L(s)  = 1  − i·2-s + (0.286 − 0.957i)3-s − 4-s + (−0.448 + 0.893i)5-s + (−0.957 − 0.286i)6-s + (−0.835 + 0.549i)7-s + i·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 + (−0.448 + 0.893i)13-s + (0.549 + 0.835i)14-s + (0.727 + 0.686i)15-s + 16-s + (−0.866 − 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.569 + 0.821i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.569 + 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4639426256 + 0.2428334521i\)
\(L(\frac12)\) \(\approx\) \(0.4639426256 + 0.2428334521i\)
\(L(1)\) \(\approx\) \(0.6100473663 - 0.3023268252i\)
\(L(1)\) \(\approx\) \(0.6100473663 - 0.3023268252i\)

Euler product

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

−17.76770407899302202390743641594, −17.273614598498568975085004846328, −16.75063365045150370015295821180, −15.89385956641999284936981649719, −15.53832533639614876900309788844, −15.337443828091049600978867576680, −14.10073587289227104157435028990, −13.49641328482685283605195085398, −12.996078298611402626395076037960, −12.29866819279968402743166826794, −11.05652338578641396675832369220, −10.39871099962609159539651647956, −9.72505619509987542150556698382, −9.06967888530185960260150206913, −8.36726371984301893842031572150, −7.89587143210767870357140054800, −7.082272981507303422815321749706, −5.997058821626102440586135653104, −5.478311344603274051893515940907, −4.6039402176396735690468788816, −4.183459835351035051079810144077, −3.36609630053739343602075859390, −2.52493207788148675663905281001, −0.74158055161011898377886138841, −0.15493590775650819409995508384, 0.61533814094493041981237519309, 2.0652771855171674933104183566, 2.401242498906643141355058970509, 2.95108573354968013358440849319, 3.89367164275082133795810161062, 4.67333662132568683403986628356, 5.80937532892138007057280389016, 6.57048669006458092815236525671, 7.14534519400897517089337671793, 8.121321748098325157839186237159, 8.60794058628801032741063106264, 9.56165986290264915521846342053, 10.11173505966329515413371875048, 10.984597890516079788142259377704, 11.63314263128107245340445379299, 12.31607304113185418058663859503, 12.74196352651096931533606258910, 13.44320829590327960619062633776, 14.26698358710462588625757752816, 14.67428815561051843844884271067, 15.556003831140746455742599597501, 16.40839227329949316310695776176, 17.42085984882468202233611490785, 18.13718162717345188036851768771, 18.646999215412532415285384248309

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