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Properties

Label 1-4033-4033.1025-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.998 - 0.0540i$
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  + (−0.342 + 0.939i)2-s + (−0.993 − 0.116i)3-s + (−0.766 − 0.642i)4-s + (−0.286 + 0.957i)5-s + (0.448 − 0.893i)6-s + (−0.686 − 0.727i)7-s + (0.866 − 0.5i)8-s + (0.973 + 0.230i)9-s + (−0.802 − 0.597i)10-s + (−0.802 + 0.597i)11-s + (0.686 + 0.727i)12-s + (0.727 − 0.686i)13-s + (0.918 − 0.396i)14-s + (0.396 − 0.918i)15-s + (0.173 + 0.984i)16-s + (0.984 − 0.173i)17-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)2-s + (−0.993 − 0.116i)3-s + (−0.766 − 0.642i)4-s + (−0.286 + 0.957i)5-s + (0.448 − 0.893i)6-s + (−0.686 − 0.727i)7-s + (0.866 − 0.5i)8-s + (0.973 + 0.230i)9-s + (−0.802 − 0.597i)10-s + (−0.802 + 0.597i)11-s + (0.686 + 0.727i)12-s + (0.727 − 0.686i)13-s + (0.918 − 0.396i)14-s + (0.396 − 0.918i)15-s + (0.173 + 0.984i)16-s + (0.984 − 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.998 - 0.0540i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.998 - 0.0540i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6343486429 + 0.01715683547i\)
\(L(\frac12)\) \(\approx\) \(0.6343486429 + 0.01715683547i\)
\(L(1)\) \(\approx\) \(0.4902201718 + 0.2131331197i\)
\(L(1)\) \(\approx\) \(0.4902201718 + 0.2131331197i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.342 + 0.939i)T \)
3 \( 1 + (-0.993 - 0.116i)T \)
5 \( 1 + (-0.286 + 0.957i)T \)
7 \( 1 + (-0.686 - 0.727i)T \)
11 \( 1 + (-0.802 + 0.597i)T \)
13 \( 1 + (0.727 - 0.686i)T \)
17 \( 1 + (0.984 - 0.173i)T \)
19 \( 1 + (-0.984 + 0.173i)T \)
23 \( 1 + (0.342 + 0.939i)T \)
29 \( 1 + (0.993 + 0.116i)T \)
31 \( 1 + (-0.973 - 0.230i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (0.998 + 0.0581i)T \)
53 \( 1 + (-0.230 - 0.973i)T \)
59 \( 1 + (0.918 + 0.396i)T \)
61 \( 1 + (0.835 - 0.549i)T \)
67 \( 1 + (0.957 - 0.286i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (0.597 + 0.802i)T \)
79 \( 1 + (-0.727 - 0.686i)T \)
83 \( 1 + (0.993 + 0.116i)T \)
89 \( 1 + (-0.835 + 0.549i)T \)
97 \( 1 + (-0.286 + 0.957i)T \)
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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

−18.50103381203390685686294431599, −17.73437852371114594044365220685, −16.81445131825609464730027825502, −16.42812476887337241082823278014, −16.01004574083947571500896097147, −15.08840915778075988463730032204, −13.90859153326484812170336111651, −13.03424942443954515194433094010, −12.64295608938631477465280407964, −12.21739953530188671106996964174, −11.348558023039025814938730591590, −10.907905352150938136914468113849, −10.038296272459519488143354628022, −9.4666994792707480787414693146, −8.55169279180587898461415513812, −8.28963351196070745476001050776, −7.08410804623873826555417176160, −6.11300379967701827096678706853, −5.47831629639661612599353074062, −4.71028728194796140403446198675, −4.04470155431321175514983463634, −3.208716440508914367152594070060, −2.23901920306977979424100028733, −1.22284363122917793880456409557, −0.56876642136900746066289497801, 0.26243572842461190632877088935, 0.97003908962461967428290521227, 2.18743097245594720270896902970, 3.56958984439052965766946630725, 3.99084854722104077290544682700, 5.20158022118253140936230309309, 5.63070418944279320686510491635, 6.540844148254312614254051862332, 6.964895753997042985719648138281, 7.60664610699359386423280287324, 8.17923777852054934752245539513, 9.458501002951794547627180638865, 10.20902010554238011286067323089, 10.507272350066087154219604781043, 11.08702754455765481302743232288, 12.2500758053757713553969796883, 12.93098649937574622913876665461, 13.52668362974482935623957028107, 14.31717159372894138087813551952, 15.189687607426873534237317972508, 15.72444708770852990924068907391, 16.15605101505856649053237202457, 17.057780202475356766226736040757, 17.50832844521044518268880182934, 18.16862240983396280900238986170

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