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Properties

Label 1-4033-4033.145-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.998 - 0.0548i$
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  + (−0.173 + 0.984i)2-s + (−0.0581 − 0.998i)3-s + (−0.939 − 0.342i)4-s + (−0.993 − 0.116i)5-s + (0.993 + 0.116i)6-s + (0.597 − 0.802i)7-s + (0.5 − 0.866i)8-s + (−0.993 + 0.116i)9-s + (0.286 − 0.957i)10-s + (0.286 + 0.957i)11-s + (−0.286 + 0.957i)12-s + (−0.396 − 0.918i)13-s + (0.686 + 0.727i)14-s + (−0.0581 + 0.998i)15-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.0581 − 0.998i)3-s + (−0.939 − 0.342i)4-s + (−0.993 − 0.116i)5-s + (0.993 + 0.116i)6-s + (0.597 − 0.802i)7-s + (0.5 − 0.866i)8-s + (−0.993 + 0.116i)9-s + (0.286 − 0.957i)10-s + (0.286 + 0.957i)11-s + (−0.286 + 0.957i)12-s + (−0.396 − 0.918i)13-s + (0.686 + 0.727i)14-s + (−0.0581 + 0.998i)15-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.998 - 0.0548i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.998 - 0.0548i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0007166673834 + 0.02609376207i\)
\(L(\frac12)\) \(\approx\) \(0.0007166673834 + 0.02609376207i\)
\(L(1)\) \(\approx\) \(0.6568959770 + 0.06035830832i\)
\(L(1)\) \(\approx\) \(0.6568959770 + 0.06035830832i\)

Euler product

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

−18.377542867199440113895731440773, −17.43635978881414765769735600680, −16.77115349952804169675755366917, −16.14319832086946872321543703113, −15.31269710910396820980151784153, −14.830466462548633290580983727336, −13.97424973851498329966374319377, −13.44262003203585740265716547240, −12.12380711676040658343963780701, −11.75766637980466462620124816036, −11.16950988129454114601186002500, −10.96845593112034016930053751567, −9.65198548498569405532288406120, −9.22512216271575373645645572737, −8.671693934650082861008280733485, −7.96300732208752601107597896438, −7.06408731090151376937011679824, −5.784971546125106814038085778086, −5.00714239285515834207327409873, −4.47314659794408925116216472205, −3.78397724877334349807492860929, −2.90366557513734593950012156951, −2.49182379254541659051251093145, −1.095322394760503593090066560052, −0.00983809322917129417142722397, 1.1223723183283271134845092057, 1.673870384266404297647172286815, 3.24218108914004023774900462625, 3.88617633771904875555854257556, 4.99135717580852771362354558833, 5.22135079387263988619157856934, 6.610311929157209750565712407052, 6.926847497479212456644043972364, 7.63776451582825461103130579783, 8.17163553485443418030490205499, 8.58465726840664724442169774654, 9.75905195062845451449009412374, 10.558360257432968316010468318467, 11.23836107758071298200921152411, 12.29782319676581093574929527208, 12.66943145764937410006400234284, 13.3316552491764442157080934198, 14.31931416022564613348057226983, 14.77294187153114741281759955535, 15.1952061012591472342592493078, 16.28700397601797457035629270460, 16.88267760009229632501103138576, 17.397071338098954973557826694054, 18.04594306585010225309873336668, 18.599424360218820883274595709

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