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Properties

Label 1-4033-4033.2901-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.894 - 0.447i$
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 + (0.686 + 0.727i)3-s + 4-s + (0.549 + 0.835i)5-s + (−0.686 − 0.727i)6-s + (−0.0581 − 0.998i)7-s − 8-s + (−0.0581 + 0.998i)9-s + (−0.549 − 0.835i)10-s + (0.549 − 0.835i)11-s + (0.686 + 0.727i)12-s + (0.835 − 0.549i)13-s + (0.0581 + 0.998i)14-s + (−0.230 + 0.973i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (0.686 + 0.727i)3-s + 4-s + (0.549 + 0.835i)5-s + (−0.686 − 0.727i)6-s + (−0.0581 − 0.998i)7-s − 8-s + (−0.0581 + 0.998i)9-s + (−0.549 − 0.835i)10-s + (0.549 − 0.835i)11-s + (0.686 + 0.727i)12-s + (0.835 − 0.549i)13-s + (0.0581 + 0.998i)14-s + (−0.230 + 0.973i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.894 - 0.447i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04998028989 + 0.2113933266i\)
\(L(\frac12)\) \(\approx\) \(-0.04998028989 + 0.2113933266i\)
\(L(1)\) \(\approx\) \(0.7454350112 + 0.2208434227i\)
\(L(1)\) \(\approx\) \(0.7454350112 + 0.2208434227i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.686 + 0.727i)T \)
5 \( 1 + (0.549 + 0.835i)T \)
7 \( 1 + (-0.0581 - 0.998i)T \)
11 \( 1 + (0.549 - 0.835i)T \)
13 \( 1 + (0.835 - 0.549i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.998 + 0.0581i)T \)
31 \( 1 + (-0.918 - 0.396i)T \)
41 \( 1 + (-0.342 + 0.939i)T \)
43 \( 1 + (0.642 + 0.766i)T \)
47 \( 1 + (-0.727 - 0.686i)T \)
53 \( 1 + (-0.957 - 0.286i)T \)
59 \( 1 + (-0.973 - 0.230i)T \)
61 \( 1 + (-0.116 + 0.993i)T \)
67 \( 1 + (-0.727 - 0.686i)T \)
71 \( 1 - T \)
73 \( 1 + (0.0581 - 0.998i)T \)
79 \( 1 + (-0.973 - 0.230i)T \)
83 \( 1 + (-0.597 + 0.802i)T \)
89 \( 1 + (-0.448 + 0.893i)T \)
97 \( 1 + (0.918 - 0.396i)T \)
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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

−18.36722210941826534674773600895, −17.48044938613031061956372710540, −16.97767396034768148333745639504, −15.95770972464213260703224243930, −15.64983401545843780350959836742, −14.573489900339945112847466499450, −14.18799534485439771750538424035, −13.061086333400372391451965507246, −12.53945682062388739036346755105, −11.98282375144461739062961728661, −11.32402822330131630039173092387, −10.17109610410285604850358055306, −9.40276321786611943404472322906, −8.963428712463175639777426461086, −8.61667958766395044266558709450, −7.74151358216450206227529051151, −6.97775061834050577840671921021, −6.17034726908841615416451181917, −5.7777337722987594811183721995, −4.48130974453318351388644908445, −3.51855080560686191454512088914, −2.42063064834249546766188792476, −1.819789802521870031522634351993, −1.50458513043540764627631753687, −0.0656807288074983575424538379, 1.44792628959034451923699674525, 2.039393361573903341914693393994, 3.18821608019450880876607583539, 3.49907119246579010916406852815, 4.386837050278211995701649370268, 5.85399875219295998573707400017, 6.20774281920827134756650700614, 7.14014434250314752103769193960, 7.86732801869378311005948039986, 8.50040292292375503783047795793, 9.24188419400397418810173998189, 9.84689034116453105114105136607, 10.6030405973505250939072136228, 10.947326341678168309425646656059, 11.41069829912891314856159169431, 13.010195608457602292909179107946, 13.43239606437545252968394726603, 14.30497284691033743802095329473, 14.87393846310751791124508187391, 15.45867136776712201762660765309, 16.329861389670344373557699881497, 16.79238208507063935098432619309, 17.4988762424314050083421832715, 18.15542522736039590638236517614, 18.91873335102046337355294362803

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