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Properties

Label 1-4033-4033.4016-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.647 + 0.761i$
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 + (0.5 − 0.866i)3-s + (−0.939 − 0.342i)4-s + (−0.866 + 0.5i)5-s + (0.766 + 0.642i)6-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (−0.342 − 0.939i)10-s + (0.342 − 0.939i)11-s + (−0.766 + 0.642i)12-s + (0.5 − 0.866i)13-s + (0.939 − 0.342i)14-s + i·15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.5 − 0.866i)3-s + (−0.939 − 0.342i)4-s + (−0.866 + 0.5i)5-s + (0.766 + 0.642i)6-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (−0.342 − 0.939i)10-s + (0.342 − 0.939i)11-s + (−0.766 + 0.642i)12-s + (0.5 − 0.866i)13-s + (0.939 − 0.342i)14-s + i·15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.647 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.647 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.647 + 0.761i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (4016, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.647 + 0.761i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01838507212 + 0.03976161453i\)
\(L(\frac12)\) \(\approx\) \(0.01838507212 + 0.03976161453i\)
\(L(1)\) \(\approx\) \(0.6765828503 + 0.02181627156i\)
\(L(1)\) \(\approx\) \(0.6765828503 + 0.02181627156i\)

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 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.342 - 0.939i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.642 - 0.766i)T \)
31 \( 1 + (-0.342 + 0.939i)T \)
41 \( 1 + (-0.642 - 0.766i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.342 - 0.939i)T \)
53 \( 1 + (-0.984 + 0.173i)T \)
59 \( 1 - T \)
61 \( 1 - iT \)
67 \( 1 + (0.984 + 0.173i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (-0.173 - 0.984i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (-0.766 + 0.642i)T \)
89 \( 1 + (0.642 + 0.766i)T \)
97 \( 1 + (0.642 - 0.766i)T \)
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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

−18.549413954350487814748524589440, −17.60573544683967818195476315031, −16.77515278779204738106882508692, −16.15976942996542640716984621074, −15.51964765930784893900596213384, −14.8485474663098997786628447187, −14.1900591366186282310885854403, −13.144935792384022613011910134886, −12.75815545945829833475866201782, −11.84362968318954361003937107271, −11.41387912407188196509457681106, −10.69598348817201049508140968498, −9.76202989724824426914742971308, −9.248032114712851379668977363600, −8.68050810320568994421262339407, −8.279618738582500275245182631011, −7.15437667829115375540984430703, −6.17247591889940972182244304241, −4.849520609973428224871028479207, −4.55938417650954019873903659115, −3.95375533904862356355046863118, −3.06132851491937628581813910557, −2.368131872689246599003437176785, −1.59824489166614136671777650634, −0.01677351426711386975114661805, 0.7501715076297082779798368947, 1.806111085373152459324499747343, 3.313496959009256040892831742810, 3.54612378481283109079373492540, 4.35063810763887661499185472150, 5.64533944445968753048063138506, 6.42047224030388206778261670163, 6.74836052635868195450933007134, 7.546549022377366751642665697, 8.28200165185088260088980600746, 8.46787949728218115558311150239, 9.51244176008613682849037074151, 10.482158235358612037673225299685, 10.97282959721355677203563004756, 12.07208798800594918709136425956, 12.76004205868504575942025915516, 13.62569778298890071865822408438, 13.81459965275122295948105349699, 14.64847666154191363180582417992, 15.44865400578069448958782987739, 15.78101299479278368803045155486, 16.75509070181256987398741801404, 17.3313204808317267944998717987, 18.0380348609571576419616612326, 18.73922165486958687469469292380

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