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Properties

Label 1-4033-4033.1266-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.248 - 0.968i$
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 + (0.835 − 0.549i)3-s + (−0.939 + 0.342i)4-s + (0.116 − 0.993i)5-s + (0.686 + 0.727i)6-s + (0.597 − 0.802i)7-s + (−0.5 − 0.866i)8-s + (0.396 − 0.918i)9-s + (0.998 − 0.0581i)10-s + (−0.998 − 0.0581i)11-s + (−0.597 + 0.802i)12-s + (0.597 − 0.802i)13-s + (0.893 + 0.448i)14-s + (−0.448 − 0.893i)15-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (0.835 − 0.549i)3-s + (−0.939 + 0.342i)4-s + (0.116 − 0.993i)5-s + (0.686 + 0.727i)6-s + (0.597 − 0.802i)7-s + (−0.5 − 0.866i)8-s + (0.396 − 0.918i)9-s + (0.998 − 0.0581i)10-s + (−0.998 − 0.0581i)11-s + (−0.597 + 0.802i)12-s + (0.597 − 0.802i)13-s + (0.893 + 0.448i)14-s + (−0.448 − 0.893i)15-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.248 - 0.968i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1266, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.248 - 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.233026249 - 1.589740402i\)
\(L(\frac12)\) \(\approx\) \(1.233026249 - 1.589740402i\)
\(L(1)\) \(\approx\) \(1.347990645 - 0.2113206100i\)
\(L(1)\) \(\approx\) \(1.347990645 - 0.2113206100i\)

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 \( 1 + (0.835 - 0.549i)T \)
5 \( 1 + (0.116 - 0.993i)T \)
7 \( 1 + (0.597 - 0.802i)T \)
11 \( 1 + (-0.998 - 0.0581i)T \)
13 \( 1 + (0.597 - 0.802i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (-0.549 - 0.835i)T \)
31 \( 1 + (0.918 + 0.396i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.642 - 0.766i)T \)
47 \( 1 + (0.957 - 0.286i)T \)
53 \( 1 + (-0.918 + 0.396i)T \)
59 \( 1 + (0.893 - 0.448i)T \)
61 \( 1 + (0.230 + 0.973i)T \)
67 \( 1 + (-0.116 - 0.993i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (0.0581 - 0.998i)T \)
79 \( 1 + (0.597 + 0.802i)T \)
83 \( 1 + (0.835 - 0.549i)T \)
89 \( 1 + (-0.230 - 0.973i)T \)
97 \( 1 + (0.116 - 0.993i)T \)
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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

−18.82790745429122802457671102186, −18.18865321961583600905446808750, −17.84321270411124458555081037659, −16.43538568261392640393144385625, −15.76013459177636572299434498153, −14.99515119648286461219184073673, −14.53196794482009010783094914943, −13.866517551494783908972683687123, −13.39615060833996685595205814731, −12.49316539702318508200006617768, −11.53495781615244673576586277543, −11.10789772155635302297450738261, −10.45090152601655703441728203379, −9.71770183549783145752172269207, −9.18074505712019525887272821345, −8.31762666482566455312302078312, −7.86448496712793564003188989598, −6.71193692704528661469769738584, −5.72729359004087909449401894228, −4.93999295707572839925206857661, −4.309416594781351450821606318844, −3.399725928823650108120981679304, −2.703576773949897853439733787093, −2.280833419342691007929871423375, −1.4781896651266599320657153274, 0.470930821226979299848975033617, 1.2297680064992277022487694747, 2.27127658471034416102669009932, 3.44135074091469093386690612898, 4.02081003831125910117740772048, 4.89288021454001765124152549984, 5.52256538490231913530074267198, 6.34207236520006610981494579527, 7.381279076223104975949539159129, 7.721805419300014430862668956467, 8.40693042457926576456794011999, 8.8486782960828229393003722733, 9.74020431486181453548553190497, 10.46155322726051274401956015436, 11.605972112842823196321537890827, 12.487868420353892545990174728981, 13.22088220717411506087340998107, 13.54494837798544516717538507845, 13.89673097051223295712334252922, 15.03887756367263847370107105036, 15.58168568743864844864128061112, 15.93557440700726594091981937772, 17.113305761250878366532008794953, 17.54145416582070015492429111224, 17.98313617772389387817417653298

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