Properties

Label 1-4033-4033.1718-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.164 + 0.986i$
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  + (0.939 + 0.342i)2-s + (−0.835 − 0.549i)3-s + (0.766 + 0.642i)4-s + (0.396 − 0.918i)5-s + (−0.597 − 0.802i)6-s + (−0.993 + 0.116i)7-s + (0.5 + 0.866i)8-s + (0.396 + 0.918i)9-s + (0.686 − 0.727i)10-s + (0.686 + 0.727i)11-s + (−0.286 − 0.957i)12-s + (−0.597 − 0.802i)13-s + (−0.973 − 0.230i)14-s + (−0.835 + 0.549i)15-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (−0.835 − 0.549i)3-s + (0.766 + 0.642i)4-s + (0.396 − 0.918i)5-s + (−0.597 − 0.802i)6-s + (−0.993 + 0.116i)7-s + (0.5 + 0.866i)8-s + (0.396 + 0.918i)9-s + (0.686 − 0.727i)10-s + (0.686 + 0.727i)11-s + (−0.286 − 0.957i)12-s + (−0.597 − 0.802i)13-s + (−0.973 − 0.230i)14-s + (−0.835 + 0.549i)15-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.164 + 0.986i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1718, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.164 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.402207833 + 1.187354000i\)
\(L(\frac12)\) \(\approx\) \(1.402207833 + 1.187354000i\)
\(L(1)\) \(\approx\) \(1.334938508 + 0.1769293373i\)
\(L(1)\) \(\approx\) \(1.334938508 + 0.1769293373i\)

Euler product

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

−18.62705287607334448737139086272, −17.40059089125194463493012720307, −16.60528208401868650030904440770, −16.53064693280083484977012336296, −15.34826684355426044180128390817, −14.94791607009923333156560490632, −14.24499733549466736945487857484, −13.528925663351713692093407815080, −12.852334910587268876464699969659, −11.94995606947061048205762732340, −11.56795102791651217244783408994, −10.83476415456488307356608208507, −10.096190522514851868578265976, −9.76991002739347326311304399449, −8.914434509053916336071403209214, −7.29506814563063276866966822938, −6.75116328267276989211457388872, −6.20762087067724454607369416826, −5.71407938955914695590327611528, −4.76040291880025758528380528835, −3.96010610875271612587943422434, −3.35755566816303432156607790122, −2.67689218249485226049206904206, −1.62198777289039865206172671558, −0.41906700404911947627225566162, 1.08813689853448213617457615715, 1.86095920047380046027420543816, 2.81234875117548098623457069815, 3.753297658284301691724489805905, 4.67398492145609607358034565141, 5.256084615379626342972245204326, 5.8206213189372562896026719379, 6.625997560297582534218949479391, 7.041201338782540021695086920012, 7.98371878238072716538795608752, 8.72366489160705198642819626478, 9.87306830777731711784952784752, 10.288961775707306621399036927450, 11.41281114493742722874092041486, 12.11399597907162409381714710580, 12.66407691929045624403788623471, 12.91064413872617929590334798155, 13.558093904355402948438941341502, 14.5651114550869357626219526977, 15.217268348498468301098398329414, 15.97621890407707057960026764893, 16.79382443171462962599376970247, 17.00500753068982805662994184313, 17.52761214540986764511916774722, 18.58447017982900929421654999611

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