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Properties

Label 1-4033-4033.1435-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.273 + 0.962i$
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  + (−0.939 − 0.342i)2-s + (−0.396 − 0.918i)3-s + (0.766 + 0.642i)4-s + (−0.230 − 0.973i)5-s + (0.0581 + 0.998i)6-s + (−0.286 + 0.957i)7-s + (−0.5 − 0.866i)8-s + (−0.686 + 0.727i)9-s + (−0.116 + 0.993i)10-s + (0.116 + 0.993i)11-s + (0.286 − 0.957i)12-s + (−0.286 + 0.957i)13-s + (0.597 − 0.802i)14-s + (−0.802 + 0.597i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (−0.396 − 0.918i)3-s + (0.766 + 0.642i)4-s + (−0.230 − 0.973i)5-s + (0.0581 + 0.998i)6-s + (−0.286 + 0.957i)7-s + (−0.5 − 0.866i)8-s + (−0.686 + 0.727i)9-s + (−0.116 + 0.993i)10-s + (0.116 + 0.993i)11-s + (0.286 − 0.957i)12-s + (−0.286 + 0.957i)13-s + (0.597 − 0.802i)14-s + (−0.802 + 0.597i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.273 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.273 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.273 + 0.962i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1435, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.273 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3656796075 + 0.2763355984i\)
\(L(\frac12)\) \(\approx\) \(0.3656796075 + 0.2763355984i\)
\(L(1)\) \(\approx\) \(0.5227291373 - 0.1167377346i\)
\(L(1)\) \(\approx\) \(0.5227291373 - 0.1167377346i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (-0.396 - 0.918i)T \)
5 \( 1 + (-0.230 - 0.973i)T \)
7 \( 1 + (-0.286 + 0.957i)T \)
11 \( 1 + (0.116 + 0.993i)T \)
13 \( 1 + (-0.286 + 0.957i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.918 - 0.396i)T \)
31 \( 1 + (0.727 + 0.686i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.984 + 0.173i)T \)
47 \( 1 + (-0.549 + 0.835i)T \)
53 \( 1 + (-0.727 + 0.686i)T \)
59 \( 1 + (0.597 + 0.802i)T \)
61 \( 1 + (0.448 - 0.893i)T \)
67 \( 1 + (0.230 - 0.973i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (0.993 - 0.116i)T \)
79 \( 1 + (-0.286 - 0.957i)T \)
83 \( 1 + (-0.396 - 0.918i)T \)
89 \( 1 + (-0.448 + 0.893i)T \)
97 \( 1 + (-0.230 - 0.973i)T \)
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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

−18.13472344908971595018787136340, −17.40996714879986154658257871569, −17.1452715753682671093915250473, −16.34931070537929251092863767393, −15.568276122359974690276339470829, −15.28491277064353521550828405449, −14.48841405281466633775746062430, −13.841506718740575545549759066501, −12.861492639757211173559938606035, −11.57349526621817497891554880875, −11.23340575783578218741133270009, −10.527153154911850962181631596672, −10.2080547646621492858314254311, −9.50597619538519099407150225522, −8.50161671024968497826571333734, −8.039272105741641884936012527981, −6.8511974787231561290421642735, −6.69444505957973388431807194157, −5.72311707807862274375244710791, −5.032494742708156388883405219675, −3.86017968535000115305681587486, −3.23111282783144490804860216022, −2.5611832704508540909235900171, −1.01090625430294278939440680756, −0.24761912636876306864377387894, 1.00271433306684376599468152435, 1.69177904196854004969890057760, 2.38604859650340638023189707917, 3.19541921482602087945877951839, 4.588560753128356241393872700282, 5.08673234611696406484103228477, 6.30483749203354690612143329017, 6.68549548084056497779142383871, 7.60378719240210452493473715521, 8.1825421308664157699979816076, 8.91847163479675550237637670076, 9.47603812659008093160433517059, 10.146736940327523342372967085384, 11.424584325009003907693413474433, 11.71077432196680675486419828299, 12.454985682454130597349681586742, 12.610791112163499867701224871846, 13.59655946804254273217183606204, 14.55416332216390225171262481054, 15.54373734776421250786447409659, 16.13349237836789851073618287646, 16.76416566064319535665271051843, 17.28546960797868671644606955657, 18.100214592301023313878967906068, 18.50105943814779104816957470892

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