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Properties

Label 1-4033-4033.2009-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.945 + 0.324i$
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 + (0.597 − 0.802i)3-s + (−0.766 − 0.642i)4-s + (0.686 + 0.727i)5-s + (−0.549 − 0.835i)6-s + (0.973 − 0.230i)7-s + (−0.866 + 0.5i)8-s + (−0.286 − 0.957i)9-s + (0.918 − 0.396i)10-s + (0.918 + 0.396i)11-s + (−0.973 + 0.230i)12-s + (−0.230 − 0.973i)13-s + (0.116 − 0.993i)14-s + (0.993 − 0.116i)15-s + (0.173 + 0.984i)16-s + (−0.984 + 0.173i)17-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s + (0.597 − 0.802i)3-s + (−0.766 − 0.642i)4-s + (0.686 + 0.727i)5-s + (−0.549 − 0.835i)6-s + (0.973 − 0.230i)7-s + (−0.866 + 0.5i)8-s + (−0.286 − 0.957i)9-s + (0.918 − 0.396i)10-s + (0.918 + 0.396i)11-s + (−0.973 + 0.230i)12-s + (−0.230 − 0.973i)13-s + (0.116 − 0.993i)14-s + (0.993 − 0.116i)15-s + (0.173 + 0.984i)16-s + (−0.984 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.945 + 0.324i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.945 + 0.324i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6856642510 - 4.104994817i\)
\(L(\frac12)\) \(\approx\) \(-0.6856642510 - 4.104994817i\)
\(L(1)\) \(\approx\) \(1.144974202 - 1.371202838i\)
\(L(1)\) \(\approx\) \(1.144974202 - 1.371202838i\)

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 \( 1 + (0.597 - 0.802i)T \)
5 \( 1 + (0.686 + 0.727i)T \)
7 \( 1 + (0.973 - 0.230i)T \)
11 \( 1 + (0.918 + 0.396i)T \)
13 \( 1 + (-0.230 - 0.973i)T \)
17 \( 1 + (-0.984 + 0.173i)T \)
19 \( 1 + (0.984 - 0.173i)T \)
23 \( 1 + (-0.342 - 0.939i)T \)
29 \( 1 + (0.597 - 0.802i)T \)
31 \( 1 + (-0.286 - 0.957i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.448 - 0.893i)T \)
53 \( 1 + (0.957 + 0.286i)T \)
59 \( 1 + (0.116 + 0.993i)T \)
61 \( 1 + (-0.0581 - 0.998i)T \)
67 \( 1 + (-0.727 - 0.686i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (0.396 - 0.918i)T \)
79 \( 1 + (0.230 - 0.973i)T \)
83 \( 1 + (-0.597 + 0.802i)T \)
89 \( 1 + (0.0581 + 0.998i)T \)
97 \( 1 + (0.686 + 0.727i)T \)
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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

−18.38578517017369129513686066632, −17.75446863079800317053689047805, −17.16040428537737594911454063857, −16.47418308306524452762150057493, −16.004706684032344503493514473401, −15.31388541439723764496782130753, −14.36446297974764341778380101527, −14.11731095548332263536313119874, −13.67569791104006141609010489002, −12.73111555032053397518683476051, −11.77871655370642378317573857960, −11.302378752727641736649565392367, −10.03136071597426498771908873988, −9.40271967875805192704508950807, −8.74228929832039939778785912363, −8.56172540005033089790266881916, −7.51438509923384155160647111560, −6.746525891213494291743348110641, −5.75690805606728222016096401351, −5.16503879229474879741360277700, −4.56373683428054867986705909381, −3.99176247040134823721892942956, −3.042142398533541685496032995049, −1.97453611160556820643553111421, −1.16352195994503737254431405747, 0.46898574394103084779752879482, 1.22519181044223018648956756996, 2.138432766619237537849656943094, 2.408319072667284009835567188668, 3.39693987483040876568489973496, 4.14392352876390342875895195438, 5.04175900821099390764708784657, 5.95478027642797322670497262923, 6.57873566902042664243987879313, 7.46592990857569074767635633299, 8.20499192312263492169798958198, 9.039597572708900279463803246, 9.64958638070439566490298351955, 10.429541676795113036449116396691, 11.103489676588388537538538429108, 11.840727952366399282326733873731, 12.36263529531679099055427801566, 13.4131444496965070714996659621, 13.60699160176068015775569166552, 14.40190988055053669426790194685, 14.85962206449616670461576327075, 15.358147108092284410886021462134, 17.051644268807581831260228394466, 17.6111515423319859074032442754, 18.09885268927215416972078728301

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