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Properties

Label 1-4033-4033.171-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.537 - 0.843i$
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  + (0.766 − 0.642i)2-s + (0.993 + 0.116i)3-s + (0.173 − 0.984i)4-s + (−0.957 − 0.286i)5-s + (0.835 − 0.549i)6-s + (−0.686 − 0.727i)7-s + (−0.5 − 0.866i)8-s + (0.973 + 0.230i)9-s + (−0.918 + 0.396i)10-s + (0.918 + 0.396i)11-s + (0.286 − 0.957i)12-s + (−0.686 − 0.727i)13-s + (−0.993 − 0.116i)14-s + (−0.918 − 0.396i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.993 + 0.116i)3-s + (0.173 − 0.984i)4-s + (−0.957 − 0.286i)5-s + (0.835 − 0.549i)6-s + (−0.686 − 0.727i)7-s + (−0.5 − 0.866i)8-s + (0.973 + 0.230i)9-s + (−0.918 + 0.396i)10-s + (0.918 + 0.396i)11-s + (0.286 − 0.957i)12-s + (−0.686 − 0.727i)13-s + (−0.993 − 0.116i)14-s + (−0.918 − 0.396i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.537 - 0.843i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.537 - 0.843i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.716869045 - 3.131706546i\)
\(L(\frac12)\) \(\approx\) \(1.716869045 - 3.131706546i\)
\(L(1)\) \(\approx\) \(1.630289422 - 1.139287631i\)
\(L(1)\) \(\approx\) \(1.630289422 - 1.139287631i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 + (0.993 + 0.116i)T \)
5 \( 1 + (-0.957 - 0.286i)T \)
7 \( 1 + (-0.686 - 0.727i)T \)
11 \( 1 + (0.918 + 0.396i)T \)
13 \( 1 + (-0.686 - 0.727i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.802 + 0.597i)T \)
31 \( 1 + (0.957 - 0.286i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (0.448 + 0.893i)T \)
53 \( 1 + (0.727 - 0.686i)T \)
59 \( 1 + (0.396 - 0.918i)T \)
61 \( 1 + (-0.549 - 0.835i)T \)
67 \( 1 + (-0.230 + 0.973i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (-0.396 + 0.918i)T \)
79 \( 1 + (-0.286 - 0.957i)T \)
83 \( 1 + (-0.396 - 0.918i)T \)
89 \( 1 + (0.448 - 0.893i)T \)
97 \( 1 + (0.727 - 0.686i)T \)
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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

−18.903022672733885912545165893038, −18.18623358000543341479665982163, −17.02152039031315432157358773283, −16.38606677256479948738338796955, −15.83877057073620809756659262446, −15.13010029417669092474233212377, −14.737360289498749327761156248, −13.95207130796774961299898870184, −13.58175863630150559103978828910, −12.40310327417961613973501661229, −12.049170580508535726387340089055, −11.675426815215973047130706369462, −10.315078251403643058175779769198, −9.39694759217762324915298767864, −8.83060416987694522486340859051, −8.16162932509298248314648124900, −7.3365179332463468442462800899, −6.94954702616271287758461881240, −6.15898186771510015156625292085, −5.20730907647516377514798012642, −4.267800379846545479128946962749, −3.68543215567462632900294776798, −2.99657856970511499588926225860, −2.53802902636397547313626532051, −1.15156481503207555688737510018, 0.78349071858028545249300879628, 1.32913941611944266693977237688, 2.72443173576225435771309726053, 3.16833809777751621360801101505, 3.78408158783473128170011398341, 4.53833912345625140196469657428, 5.03015373220284425204373768475, 6.34063311678093469055905105458, 7.1525786986962416294187842417, 7.56082138931708787932536994029, 8.63899601573484970858632944691, 9.3542349697841353854372036103, 10.04641640040728737739990853099, 10.52892933514596680662249981729, 11.56349519766828697641885193909, 12.2242410191405037589338197130, 12.82547310155601521779456364247, 13.309589159555279750739664975339, 14.35385134537497442980805798742, 14.53241731894845494615655196411, 15.45423858136283829591703939295, 15.84282827793766089104411323316, 16.700750620173858589080814144856, 17.54672480203494440810031730796, 18.77467082025963718842045053047

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