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Properties

Label 1-4033-4033.512-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.0631 - 0.998i$
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  + 2-s − 3-s + 4-s − i·5-s − 6-s + 7-s + 8-s + 9-s − i·10-s − i·11-s − 12-s + 13-s + 14-s + i·15-s + 16-s − 17-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − i·5-s − 6-s + 7-s + 8-s + 9-s − i·10-s − i·11-s − 12-s + 13-s + 14-s + i·15-s + 16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0631 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0631 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.0631 - 0.998i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (512, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.0631 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.090951501 - 2.227356169i\)
\(L(\frac12)\) \(\approx\) \(2.090951501 - 2.227356169i\)
\(L(1)\) \(\approx\) \(1.655476337 - 0.5858910734i\)
\(L(1)\) \(\approx\) \(1.655476337 - 0.5858910734i\)

Euler product

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

−18.39521685627135083846154917927, −17.79194477584855258982310407463, −17.573216707617021727809510462924, −16.38356262402991772655096743292, −15.69979743004202036728939475468, −15.37073731939690857104561459922, −14.45053904878409342970848149148, −13.95981141816212744898136782857, −13.219783268447352156622105177, −12.32693279527212748366985436867, −11.7451177949176194641361528461, −11.2639137459887776572703576426, −10.5079037367672080477457973451, −10.26440088825422401686682233109, −8.929224432752637193030932203412, −7.74879695610639612231990318036, −7.145438752821256737728885708865, −6.69704337325345000424908825455, −5.77272386393050678581117600837, −5.31343051689829365750993668233, −4.34282790602295059830130902380, −3.96331198823969572668670135799, −2.852578046200331793117522116521, −1.88340977029374999681945582185, −1.34854350304978191068700401155, 0.68181171960109391051471941791, 1.44696028133042260177035839714, 2.18943754287497879492942966119, 3.57560807627112439281188992435, 4.246572437495942049960568183229, 4.788171162490938795338804752423, 5.60655808577070738235035309983, 5.94951150554298837252283068374, 6.74544550247538582408092476727, 7.93182416734822488138260461328, 8.180920911989267487080921732042, 9.36201751806489152348444284150, 10.26944967193465721966092210522, 11.212085828683039393584760547686, 11.491391492023354261268363456610, 11.92240871332036789302336636406, 12.96835017551461108968004897061, 13.43722392738248818565177951193, 13.90002021406555345930244013980, 14.99524891519843040692277068292, 15.717969313110080464064153994640, 16.22212061577699381219917777224, 16.6786212707197648968587911009, 17.549237670622392320432605020884, 18.07533966261303419380283920422

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