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Properties

Label 1-4033-4033.1190-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.976 - 0.214i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (0.642 + 0.766i)2-s + (−0.893 − 0.448i)3-s + (−0.173 + 0.984i)4-s + (−0.918 + 0.396i)5-s + (−0.230 − 0.973i)6-s + (−0.993 − 0.116i)7-s + (−0.866 + 0.5i)8-s + (0.597 + 0.802i)9-s + (−0.893 − 0.448i)10-s + (0.893 − 0.448i)11-s + (0.597 − 0.802i)12-s + (−0.116 + 0.993i)13-s + (−0.549 − 0.835i)14-s + (0.998 + 0.0581i)15-s + (−0.939 − 0.342i)16-s + (−0.342 + 0.939i)17-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (−0.893 − 0.448i)3-s + (−0.173 + 0.984i)4-s + (−0.918 + 0.396i)5-s + (−0.230 − 0.973i)6-s + (−0.993 − 0.116i)7-s + (−0.866 + 0.5i)8-s + (0.597 + 0.802i)9-s + (−0.893 − 0.448i)10-s + (0.893 − 0.448i)11-s + (0.597 − 0.802i)12-s + (−0.116 + 0.993i)13-s + (−0.549 − 0.835i)14-s + (0.998 + 0.0581i)15-s + (−0.939 − 0.342i)16-s + (−0.342 + 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.976 - 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.976 - 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.976 - 0.214i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.976 - 0.214i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09473301757 + 0.01029616638i\)
\(L(\frac12)\) \(\approx\) \(0.09473301757 + 0.01029616638i\)
\(L(1)\) \(\approx\) \(0.5854929715 + 0.3989467323i\)
\(L(1)\) \(\approx\) \(0.5854929715 + 0.3989467323i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.642 + 0.766i)T \)
3 \( 1 + (-0.893 - 0.448i)T \)
5 \( 1 + (-0.918 + 0.396i)T \)
7 \( 1 + (-0.993 - 0.116i)T \)
11 \( 1 + (0.893 - 0.448i)T \)
13 \( 1 + (-0.116 + 0.993i)T \)
17 \( 1 + (-0.342 + 0.939i)T \)
19 \( 1 + (0.984 - 0.173i)T \)
23 \( 1 + (-0.984 - 0.173i)T \)
29 \( 1 + (-0.998 + 0.0581i)T \)
31 \( 1 + (0.116 + 0.993i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.642 + 0.766i)T \)
47 \( 1 + (0.686 - 0.727i)T \)
53 \( 1 + (-0.396 - 0.918i)T \)
59 \( 1 + (0.998 + 0.0581i)T \)
61 \( 1 + (-0.727 + 0.686i)T \)
67 \( 1 + (-0.993 + 0.116i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (-0.893 + 0.448i)T \)
79 \( 1 + (0.802 - 0.597i)T \)
83 \( 1 + (-0.835 + 0.549i)T \)
89 \( 1 + (-0.230 - 0.973i)T \)
97 \( 1 + (0.802 + 0.597i)T \)
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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

−18.47747083013102579421054554716, −17.697377081050769230337346582638, −16.84843333666496271767934941831, −16.118400823587910495489240341221, −15.48286967188124400657922891670, −15.23271070796902258333085839402, −14.14162530614756776903189256743, −13.318241717323064009042241590014, −12.54299163650034082198872237788, −12.10181255033378528150445909687, −11.66775624983932350915046552554, −10.9037374447420650243201926546, −10.15089779984289683719210205250, −9.41109372390749640202513664547, −9.12123720984174306421249123293, −7.62696774892896619555813240467, −6.96820952634736634664247283284, −6.02125101340342485186046189906, −5.50505700671383786881967897596, −4.7089828450006731470810924818, −3.90508716583646349358135419279, −3.56978907983642374289038036315, −2.59721302322791308954585290149, −1.314505467659902466245660972401, −0.461517757536513389975835006753, 0.027267563830390513208124359570, 1.2693248520158756662670038117, 2.51472689459159546080244133723, 3.59129679132829482250960619316, 3.975653608999067318486151700389, 4.77744561790866549595411197824, 5.82930983238305460122898301047, 6.356424877498956090821727967656, 6.90802279973153055685788275485, 7.406913311978162664191335956916, 8.29905538319256667337102320420, 9.08175241039954327966601914392, 10.030472893252139785043019724, 11.019447456384295296431086828, 11.71525070668704277074305832426, 12.043129932182150144882883654728, 12.80901165138249344681634848287, 13.48720873487805884001482534574, 14.200331278666999991316389303716, 14.872023969521686438954813942673, 15.835965591693025070012551257365, 16.210727394907669096325725587062, 16.67409409440792605026363387714, 17.392811624589842128279939043254, 18.22490733284308599595835649188

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