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Properties

Label 1-4033-4033.323-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.578 + 0.815i$
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  + (0.5 − 0.866i)2-s + (0.173 + 0.984i)3-s + (−0.5 − 0.866i)4-s + (0.939 + 0.342i)5-s + (0.939 + 0.342i)6-s + (−0.939 − 0.342i)7-s − 8-s + (−0.939 + 0.342i)9-s + (0.766 − 0.642i)10-s + (0.766 + 0.642i)11-s + (0.766 − 0.642i)12-s + (0.939 + 0.342i)13-s + (−0.766 + 0.642i)14-s + (−0.173 + 0.984i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.173 + 0.984i)3-s + (−0.5 − 0.866i)4-s + (0.939 + 0.342i)5-s + (0.939 + 0.342i)6-s + (−0.939 − 0.342i)7-s − 8-s + (−0.939 + 0.342i)9-s + (0.766 − 0.642i)10-s + (0.766 + 0.642i)11-s + (0.766 − 0.642i)12-s + (0.939 + 0.342i)13-s + (−0.766 + 0.642i)14-s + (−0.173 + 0.984i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.578 + 0.815i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.578 + 0.815i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.830797337 + 0.9457108253i\)
\(L(\frac12)\) \(\approx\) \(1.830797337 + 0.9457108253i\)
\(L(1)\) \(\approx\) \(1.393280959 - 0.03239338175i\)
\(L(1)\) \(\approx\) \(1.393280959 - 0.03239338175i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (0.766 + 0.642i)T \)
13 \( 1 + (0.939 + 0.342i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 - T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.766 + 0.642i)T \)
31 \( 1 + (-0.173 + 0.984i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.939 - 0.342i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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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

−18.462786099112235660849732417786, −17.326604232517191832876701892509, −17.085582719012634610211727259597, −16.55517688713682459133011809542, −15.49659335888564803900697329394, −14.96734073078387533845986778944, −14.04262897052329415739486150306, −13.60937049909918013599378050216, −12.94917283826157697359629886711, −12.682111534908171772476513279730, −11.814583348912617140737339893039, −10.93554287250567624953753375771, −9.74801702279871655907583851898, −9.02332062075140423926974273047, −8.602018568277372284928955824236, −7.86985122695985414594323420524, −6.862216223842469886630664291617, −6.28808666290427211977191893802, −5.865354848018879966980929406307, −5.39013924905077453145756301352, −3.88073040134854214111019061521, −3.4651921855725724087955139687, −2.46694143422733580813633056561, −1.59897140231368503093467117528, −0.48133099266773884063381729198, 1.11015988445852400659225899584, 2.01553867262438027792081927370, 2.96718989828302504621240231237, 3.363876128285822772961169182053, 4.27336159462542489515680183948, 4.83337702680427355797726311074, 5.808770822735844071286288080590, 6.36274376538662937918356603621, 7.069295870693559871869560284887, 8.768918678229314130314224463171, 9.02657278319575727613667471741, 9.86850037458193925777807422621, 10.17704973520379870715209578893, 10.93009118335897932593986092382, 11.50369484043160273426919480591, 12.54700190721352716225287689108, 13.1239525047358250392218101542, 13.80588916767278686040777381497, 14.547543226677800959308585766816, 14.76520796222781991991789273215, 15.82346285854941541636208306131, 16.54870812448201506816099796454, 17.0904035537471697688636316148, 18.09605117310741207176377344374, 18.61616844709571308658742268932

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