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Properties

Label 1-4033-4033.695-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.0519 + 0.998i$
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  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.866 + 0.5i)5-s + (0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.866 + 0.5i)10-s + (0.866 + 0.5i)11-s − 12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.866 + 0.5i)5-s + (0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.866 + 0.5i)10-s + (0.866 + 0.5i)11-s − 12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0519 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0519 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.0519 + 0.998i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (695, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.0519 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.120536701 + 1.063763227i\)
\(L(\frac12)\) \(\approx\) \(1.120536701 + 1.063763227i\)
\(L(1)\) \(\approx\) \(0.9757752676 + 0.3288048127i\)
\(L(1)\) \(\approx\) \(0.9757752676 + 0.3288048127i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 - iT \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.866 + 0.5i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + iT \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−18.424640012354079093740437773550, −17.32329312668811270545003365610, −16.959226794171855043054059597457, −16.68770501394476713634124525855, −15.76470567266725228719824299952, −14.73695991942886930529493031659, −14.000076170389351036458604392225, −13.59937293058575846128606062639, −12.76379386389441980991148413010, −12.21051207436108495006385035202, −11.15773333761967149425986079136, −10.40831830487542990891894090423, −9.9943885526491243052812501060, −9.59194438183971521067098549214, −8.714505134763351700138862555833, −8.16246138450870994365397563372, −7.44207211419333985802177380169, −6.11852824469518238651567665316, −5.55134094315818785083126500438, −4.28562322867346483804471633387, −4.017629162804988625015991962352, −3.20842090843968420280697657685, −2.34491115229370398355802423652, −1.53912905465876797668198865067, −0.53994759311480281834018374295, 0.960017632914202236735408489260, 2.02020647480480776556236039668, 2.33837140259843939844107723352, 3.46311874481076447487336188244, 4.633597883653233681441011991, 5.54283570173634672386884310003, 6.27855484077349518889623152298, 6.74200579622712637997784096762, 7.22742704093980080509867481868, 8.14839870784110131042074961688, 9.085584508029104012072919108314, 9.38570979244443517947912373672, 9.840930654867191685807316604, 11.04372793519518122608712895130, 11.895104969613552719954554692409, 12.60180825640118598499237993138, 13.3883467190405836434482340775, 14.16789103940642282032868007268, 14.41337294663846291448772770239, 15.11046096580317477128119446132, 15.90421053339819557145807614481, 16.76245286575468155443747937578, 17.40898402030180433077279397180, 18.01397716874103697882715481019, 18.45168728308403987451437267799

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