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Properties

Label 1-4033-4033.189-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.615 + 0.788i$
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 + (0.396 − 0.918i)3-s + 4-s + (0.286 + 0.957i)5-s + (−0.396 + 0.918i)6-s + (−0.686 + 0.727i)7-s − 8-s + (−0.686 − 0.727i)9-s + (−0.286 − 0.957i)10-s + (−0.286 + 0.957i)11-s + (0.396 − 0.918i)12-s + (0.286 + 0.957i)13-s + (0.686 − 0.727i)14-s + (0.993 + 0.116i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (0.396 − 0.918i)3-s + 4-s + (0.286 + 0.957i)5-s + (−0.396 + 0.918i)6-s + (−0.686 + 0.727i)7-s − 8-s + (−0.686 − 0.727i)9-s + (−0.286 − 0.957i)10-s + (−0.286 + 0.957i)11-s + (0.396 − 0.918i)12-s + (0.286 + 0.957i)13-s + (0.686 − 0.727i)14-s + (0.993 + 0.116i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.615 + 0.788i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.615 + 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.139635048 + 0.5559836497i\)
\(L(\frac12)\) \(\approx\) \(1.139635048 + 0.5559836497i\)
\(L(1)\) \(\approx\) \(0.8268710644 + 0.08518544257i\)
\(L(1)\) \(\approx\) \(0.8268710644 + 0.08518544257i\)

Euler product

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

−18.20291246053535588002069068949, −17.64584320318513314666692985513, −16.83565761255185937830034688655, −16.279572748176567155673609029582, −15.947974140852336179584672747015, −15.43530323409170862222178898417, −14.227536680375164146312487647316, −13.65275878719470217635036658603, −13.00765565744667512912928110378, −11.97156642268050642005393030765, −11.29226249845422266326806765476, −10.460636464365365665649072522, −9.96755175633697075483192791919, −9.41709593585335852760707880397, −8.76215538280547822569961859413, −8.0751921046824638805572992312, −7.52758529300737931265977889784, −6.44635036525860457227607838840, −5.489328272900878404584859544491, −5.14576951823843962433764268851, −3.827212775738006322215159547851, −3.17294731873825542528968929387, −2.60902047546838309184525868472, −1.132199665198101928154039393333, −0.63596122214203270190330819978, 0.89867282112762449331822278270, 2.04214177304099749712342495690, 2.32426586810049124978247155895, 3.08509898416260502763111438240, 3.987396860041971656103416820778, 5.66108282128020969901840362264, 6.20796536426837978326486122552, 6.73754413967347169203246050868, 7.41100006438154856622685938254, 8.05507992635802151667381108474, 8.85395144370726779474983628698, 9.577874851751015897372086995031, 10.0621391585176649276764766599, 10.883814130881915661780683051381, 11.8970209521566729946088686674, 12.15410468995516107062570683454, 12.95061190971215378180486398702, 13.97265075016154819262735996459, 14.47906900935767830540123600364, 15.29773747877434597599415163691, 15.75087938693492187784831659392, 16.74931288352370409834040319243, 17.49219499913721950906558651824, 18.03620091657976268270938359094, 18.6498600767875676372878422215

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