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Properties

Label 1-4033-4033.2743-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.939 + 0.342i$
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  + (−0.766 − 0.642i)2-s + (0.686 + 0.727i)3-s + (0.173 + 0.984i)4-s + (0.998 − 0.0581i)5-s + (−0.0581 − 0.998i)6-s + (0.893 + 0.448i)7-s + (0.5 − 0.866i)8-s + (−0.0581 + 0.998i)9-s + (−0.802 − 0.597i)10-s + (0.802 − 0.597i)11-s + (−0.597 + 0.802i)12-s + (0.835 − 0.549i)13-s + (−0.396 − 0.918i)14-s + (0.727 + 0.686i)15-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.686 + 0.727i)3-s + (0.173 + 0.984i)4-s + (0.998 − 0.0581i)5-s + (−0.0581 − 0.998i)6-s + (0.893 + 0.448i)7-s + (0.5 − 0.866i)8-s + (−0.0581 + 0.998i)9-s + (−0.802 − 0.597i)10-s + (0.802 − 0.597i)11-s + (−0.597 + 0.802i)12-s + (0.835 − 0.549i)13-s + (−0.396 − 0.918i)14-s + (0.727 + 0.686i)15-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.939 + 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.677111479 + 0.4721046766i\)
\(L(\frac12)\) \(\approx\) \(2.677111479 + 0.4721046766i\)
\(L(1)\) \(\approx\) \(1.442172030 + 0.07751172516i\)
\(L(1)\) \(\approx\) \(1.442172030 + 0.07751172516i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (0.686 + 0.727i)T \)
5 \( 1 + (0.998 - 0.0581i)T \)
7 \( 1 + (0.893 + 0.448i)T \)
11 \( 1 + (0.802 - 0.597i)T \)
13 \( 1 + (0.835 - 0.549i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (0.802 + 0.597i)T \)
31 \( 1 + (-0.727 - 0.686i)T \)
41 \( 1 + (-0.642 + 0.766i)T \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (-0.448 - 0.893i)T \)
53 \( 1 + (-0.802 - 0.597i)T \)
59 \( 1 + (0.686 - 0.727i)T \)
61 \( 1 + (-0.918 + 0.396i)T \)
67 \( 1 + (-0.918 - 0.396i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (0.993 - 0.116i)T \)
79 \( 1 + (0.993 - 0.116i)T \)
83 \( 1 + (-0.893 - 0.448i)T \)
89 \( 1 + (0.957 + 0.286i)T \)
97 \( 1 + (-0.957 - 0.286i)T \)
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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

−18.0153693754608234134870563676, −17.95690181165222284268496784701, −17.30752852953590471690356039743, −16.618027848683059313268553353604, −15.67626855132557187475297377367, −14.93972539222419539988107980171, −14.1186271464185602016746570970, −14.005171368917306853386469677835, −13.37843029931379352423772917985, −12.2094561214791150158377557017, −11.47006995642582189494862224022, −10.74817757713385070470615708506, −9.78586733156011862159901303492, −9.2382588357590418877469832083, −8.815401182941845308628164056475, −7.86359535089851706846414389203, −7.24839365082155391364363788479, −6.74620561322229941666614671847, −5.9773447625109321539495669192, −5.17548681989416351172473158904, −4.30139390281083207944713473232, −3.10927847435958775645152665160, −2.1351897782092174835083674748, −1.33722601228211812522813715543, −1.13207620165258499417546431770, 1.18692493948061854903685656487, 1.64301356009024651261468517978, 2.54962550555465730560982093563, 3.31365871641913741039782545002, 3.94257838405611173847403944103, 4.95473403415360844951294328446, 5.74260204280527803337527653848, 6.55346700725728162874633055021, 7.85233430940314674650471372999, 8.27394520195987403489670189658, 8.894728798905509113489150883249, 9.42091146000921283628111859543, 10.30700287607664638514930776074, 10.64018691725584337287660742874, 11.40822542789895158816102134297, 12.24986221235069816971318214377, 13.02756911247122978286468241814, 13.798190408676839122420928366442, 14.44080297117884485564998457454, 14.94796184330660918134677902348, 16.08282676076981101303443485501, 16.55241287990198224357670119269, 17.15737990331786913436143324069, 17.959593380705029643645652037864, 18.55296908960417845896227821440

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