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Properties

Label 1-4033-4033.2598-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.983 - 0.180i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.983 - 0.180i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2598, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.983 - 0.180i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7424087226 - 0.06740165239i\)
\(L(\frac12)\) \(\approx\) \(0.7424087226 - 0.06740165239i\)
\(L(1)\) \(\approx\) \(0.5904365824 - 0.07644744304i\)
\(L(1)\) \(\approx\) \(0.5904365824 - 0.07644744304i\)

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

−18.06803014026208737756522130251, −17.56083140210742986423200036538, −17.0369088248115440903658817362, −16.76043312497419786404573261209, −15.79899493572051770185987959546, −15.380124963705514004436900709312, −14.787460521078225893562873090543, −13.81709319144881377782845769262, −12.78010529344563410022389876601, −12.428138146686074938112408431177, −11.06189567669897109202533304419, −10.9349162073835494659240258781, −9.92769307440128876926197595291, −9.60097841446650041914367127739, −8.97557862160748704058211236450, −8.0941127180587992591383524788, −7.39684503395435709993604702831, −6.607955669016996524916776441898, −5.72333235224089925113296678678, −5.086913250253592706705909376913, −4.453355829760315931696780963555, −3.50866456570267601330840239697, −2.45651199904397879354401565525, −1.3222634222273716467610911962, −0.55956108790356044497992395548, 0.56004365065483042582216855544, 1.78197489239399200311936834718, 2.34397450168233327707535870549, 2.95002461254435264083798159281, 3.80327118011285329340378883819, 5.471453822148454932171268990694, 5.87259320403143894807232384372, 6.78055217785643432363196866641, 7.17038678593751193525835235864, 7.91626151143952720358971972102, 8.93557822225324278729207427953, 9.20120030933183715114882000117, 10.30781808346025878371944748832, 10.935217214025287166433575373188, 11.654178144952301785680531651502, 11.89765194175707486908981247300, 12.85789897961776748932333509347, 13.53252693237816710880027977850, 14.366615059061452700323044640882, 15.045155031686246410121901349298, 16.10676587677270632909941666179, 16.521929347321676930238613575021, 17.22512328499805995448522295317, 18.13298016615710494474821197427, 18.5128620151983060048923418443

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