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Properties

Label 1-4033-4033.117-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.706 - 0.708i$
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  + 2-s + (0.5 − 0.866i)3-s + 4-s + (−0.866 − 0.5i)5-s + (0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.866 − 0.5i)10-s + (0.866 − 0.5i)11-s + (0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.866 + 0.5i)15-s + 16-s − 17-s + ⋯
L(s)  = 1  + 2-s + (0.5 − 0.866i)3-s + 4-s + (−0.866 − 0.5i)5-s + (0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.866 − 0.5i)10-s + (0.866 − 0.5i)11-s + (0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.866 + 0.5i)15-s + 16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.706 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.706 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.706 - 0.708i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.706 - 0.708i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9734035627 - 2.344961464i\)
\(L(\frac12)\) \(\approx\) \(0.9734035627 - 2.344961464i\)
\(L(1)\) \(\approx\) \(1.607025054 - 0.7347636832i\)
\(L(1)\) \(\approx\) \(1.607025054 - 0.7347636832i\)

Euler product

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

−19.17927522483289569145888759470, −17.92816355972541879873764675108, −17.09726944478668564834352016175, −16.333633762014375415960509679455, −15.83668961596229167420533128635, −15.25108930241753792678065105055, −14.6570466471663919869750590024, −14.0794321663024915218405234673, −13.46411404803785796445718105230, −12.59854232448780723260318160684, −11.86466020162767352218010692535, −11.1661588330153302782118965433, −10.542577803056460630783061370250, −9.951095120140030573531893075412, −9.16871948000684954350401857582, −7.91158965770486209017644450037, −7.58069786184213995999306682527, −6.764480125169638563137491733958, −6.033113878312836526550032822478, −4.89800611633691064274794057032, −4.349156896533503539493518324796, −3.79644827726988542884509690258, −3.11768962478251696236291557869, −2.558157398938584005471584195887, −1.26368141850222188532986298006, 0.429629600674434826415626842935, 1.72254383999006070764293229520, 2.247071868825888910314770805013, 3.242474874041898716221911492838, 3.803503021897495103156637067779, 4.55534118348406556729252794637, 5.587131412845824514363271124590, 6.23890862063739051783795316498, 6.9131560515757656350875774541, 7.55344373169788811861248090237, 8.376663440101139581509373045618, 9.05552854492571713934894616126, 9.70321690157799675785954022383, 11.19001261389545092044801593314, 11.71427122007526838859634465170, 12.11214254506528883517094250389, 12.63453403503450082133185001673, 13.5507832043716478968787185392, 13.93771409688517782373003929381, 14.729384311793054118674710092289, 15.4915037415935200344896983189, 15.851081551447167846935086698, 16.752957781807410088932154783460, 17.36617941366942117975420073114, 18.58237593149115109566125218440

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