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Properties

Label 1-4033-4033.1537-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.892 - 0.450i$
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 + (0.993 + 0.116i)3-s + 4-s + (−0.727 + 0.686i)5-s + (−0.993 − 0.116i)6-s + (0.973 − 0.230i)7-s − 8-s + (0.973 + 0.230i)9-s + (0.727 − 0.686i)10-s + (−0.727 − 0.686i)11-s + (0.993 + 0.116i)12-s + (0.686 + 0.727i)13-s + (−0.973 + 0.230i)14-s + (−0.802 + 0.597i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (0.993 + 0.116i)3-s + 4-s + (−0.727 + 0.686i)5-s + (−0.993 − 0.116i)6-s + (0.973 − 0.230i)7-s − 8-s + (0.973 + 0.230i)9-s + (0.727 − 0.686i)10-s + (−0.727 − 0.686i)11-s + (0.993 + 0.116i)12-s + (0.686 + 0.727i)13-s + (−0.973 + 0.230i)14-s + (−0.802 + 0.597i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.892 - 0.450i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.892 - 0.450i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.595375980 - 0.3794811968i\)
\(L(\frac12)\) \(\approx\) \(1.595375980 - 0.3794811968i\)
\(L(1)\) \(\approx\) \(1.022198825 + 0.01667366007i\)
\(L(1)\) \(\approx\) \(1.022198825 + 0.01667366007i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.993 + 0.116i)T \)
5 \( 1 + (-0.727 + 0.686i)T \)
7 \( 1 + (0.973 - 0.230i)T \)
11 \( 1 + (-0.727 - 0.686i)T \)
13 \( 1 + (0.686 + 0.727i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.230 - 0.973i)T \)
31 \( 1 + (0.998 + 0.0581i)T \)
41 \( 1 + (0.984 - 0.173i)T \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (-0.116 - 0.993i)T \)
53 \( 1 + (-0.918 + 0.396i)T \)
59 \( 1 + (-0.597 - 0.802i)T \)
61 \( 1 + (0.448 - 0.893i)T \)
67 \( 1 + (-0.116 - 0.993i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.973 - 0.230i)T \)
79 \( 1 + (-0.597 - 0.802i)T \)
83 \( 1 + (0.835 - 0.549i)T \)
89 \( 1 + (-0.957 - 0.286i)T \)
97 \( 1 + (-0.998 + 0.0581i)T \)
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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

−18.43691568834578848427757465389, −17.904620343534634544121184338198, −17.527405897394371376570760866471, −16.09242346574242639772447342500, −15.98824008784132288921767395915, −15.261519278121519966399448879, −14.72334293295474347568621529382, −13.773393145371748567711048537713, −12.93704062666662728020042937849, −12.32370071868319929813705368023, −11.5029052420717343702440279546, −10.97358995836437574026940808341, −9.984537021202873564897898577807, −9.35418811543382906726605472423, −8.679822870224359428244991696843, −8.11653571470983638278499299338, −7.520813731453323915401513650685, −7.23745921248410274425678694278, −5.827809793153178300380313362750, −5.00317898019157714278226998868, −4.21372884386777488148118891478, −3.11398017998405014586004700072, −2.61928972898232895623315144289, −1.41317217260565525385868089732, −1.12181860538100295121840946348, 0.61464351296650461806800478865, 1.69174875737245803793284018103, 2.41700262249955746928104197969, 3.14945036692902681074973550236, 3.934710821364506820210217358915, 4.70856092327279723551609323893, 6.02709887120406389523636991084, 6.77216585557842939638729925327, 7.58122268041551133010820848559, 8.02294494792653315520542556408, 8.568006562801795631305848332738, 9.17467309110491012499379982511, 10.2821810716897963120866744582, 10.67846662430035522497225916806, 11.32010947491320517865136115727, 11.93995421048477893763941523757, 13.02001207236441204858584789793, 13.93022495288143953443055248015, 14.3793744356447617963682608905, 15.26565985622131255140766083394, 15.6323202656958287523692267806, 16.23229397853062672165136417496, 17.15205491680549974238004026, 17.981310467721119013769522530870, 18.58986430982770280028624729244

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