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Properties

Label 1-4033-4033.2674-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.962 - 0.271i$
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  + (−0.342 − 0.939i)2-s + (0.597 + 0.802i)3-s + (−0.766 + 0.642i)4-s + (−0.686 + 0.727i)5-s + (0.549 − 0.835i)6-s + (0.973 + 0.230i)7-s + (0.866 + 0.5i)8-s + (−0.286 + 0.957i)9-s + (0.918 + 0.396i)10-s + (0.918 − 0.396i)11-s + (−0.973 − 0.230i)12-s + (0.230 − 0.973i)13-s + (−0.116 − 0.993i)14-s + (−0.993 − 0.116i)15-s + (0.173 − 0.984i)16-s + (0.984 + 0.173i)17-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (0.597 + 0.802i)3-s + (−0.766 + 0.642i)4-s + (−0.686 + 0.727i)5-s + (0.549 − 0.835i)6-s + (0.973 + 0.230i)7-s + (0.866 + 0.5i)8-s + (−0.286 + 0.957i)9-s + (0.918 + 0.396i)10-s + (0.918 − 0.396i)11-s + (−0.973 − 0.230i)12-s + (0.230 − 0.973i)13-s + (−0.116 − 0.993i)14-s + (−0.993 − 0.116i)15-s + (0.173 − 0.984i)16-s + (0.984 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.962 - 0.271i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2674, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.962 - 0.271i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.334091198 - 0.3228369377i\)
\(L(\frac12)\) \(\approx\) \(2.334091198 - 0.3228369377i\)
\(L(1)\) \(\approx\) \(1.100223024 - 0.06209173865i\)
\(L(1)\) \(\approx\) \(1.100223024 - 0.06209173865i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.342 - 0.939i)T \)
3 \( 1 + (0.597 + 0.802i)T \)
5 \( 1 + (-0.686 + 0.727i)T \)
7 \( 1 + (0.973 + 0.230i)T \)
11 \( 1 + (0.918 - 0.396i)T \)
13 \( 1 + (0.230 - 0.973i)T \)
17 \( 1 + (0.984 + 0.173i)T \)
19 \( 1 + (-0.984 - 0.173i)T \)
23 \( 1 + (0.342 - 0.939i)T \)
29 \( 1 + (-0.597 - 0.802i)T \)
31 \( 1 + (0.286 - 0.957i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (-0.448 + 0.893i)T \)
53 \( 1 + (0.957 - 0.286i)T \)
59 \( 1 + (-0.116 + 0.993i)T \)
61 \( 1 + (0.0581 - 0.998i)T \)
67 \( 1 + (-0.727 + 0.686i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (0.396 + 0.918i)T \)
79 \( 1 + (-0.230 - 0.973i)T \)
83 \( 1 + (-0.597 - 0.802i)T \)
89 \( 1 + (-0.0581 + 0.998i)T \)
97 \( 1 + (-0.686 + 0.727i)T \)
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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

−18.31833679891426264069838839348, −17.560384505969760945684179007814, −16.86056093609740726337164266941, −16.60814740729632484123014135857, −15.37393750668585415944024425511, −15.02138605363460809534005257134, −14.167788689818561502371120726133, −13.93081582984836863121781459591, −12.96238703626752134996137587724, −12.18936296025660416382885893449, −11.681963398784850727143577293077, −10.73381948570861771641841849547, −9.62575147718378571850821995937, −8.872401056653498230249307179326, −8.60858735009222383702498574861, −7.8184433729562011360386943129, −7.16392466180409778451092588342, −6.77555266160497435946464428581, −5.645556986375719974969446637899, −4.950979036553640053817374925429, −4.02832499902892453904154221296, −3.61330502287628827412416829466, −1.817076953421552000995770148647, −1.472653552832593791534726677018, −0.665164000613148193732281509182, 0.52563189675640502116974296640, 1.531109170466465380942824048722, 2.59876726526135694839090978975, 2.994476762622501764647581424977, 4.03099252613265525187583633533, 4.22548885843199412315104286685, 5.244929116024438854179199261004, 6.22927076773873881655425349333, 7.52327704574553998187812911695, 8.1791525097586433991841827045, 8.39576356224229625839292613440, 9.36215733881999074698353799956, 10.09569896088365287848665923879, 10.75420407740667402366503319392, 11.25177779167695042317262325677, 11.76542289698878015226822691806, 12.69641569011636014133499918939, 13.50237653551660116981229226655, 14.43084408213794942014816173921, 14.73074030121115987742988663632, 15.28991511734992843912990395061, 16.394625168482997313728302732206, 16.939679272643688221149786420729, 17.691573130179789762964690192982, 18.54543749536921845053003990444

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