↑

Properties

Label 1-4033-4033.2917-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.391 - 0.920i$
Analytic cond. $433.406$
Root an. cond. $433.406$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.342 + 0.939i)2-s + (0.286 + 0.957i)3-s + (−0.766 + 0.642i)4-s + (0.998 + 0.0581i)5-s + (−0.802 + 0.597i)6-s + (0.893 − 0.448i)7-s + (−0.866 − 0.5i)8-s + (−0.835 + 0.549i)9-s + (0.286 + 0.957i)10-s + (−0.286 + 0.957i)11-s + (−0.835 − 0.549i)12-s + (0.448 + 0.893i)13-s + (0.727 + 0.686i)14-s + (0.230 + 0.973i)15-s + (0.173 − 0.984i)16-s + (0.984 + 0.173i)17-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)2-s + (0.286 + 0.957i)3-s + (−0.766 + 0.642i)4-s + (0.998 + 0.0581i)5-s + (−0.802 + 0.597i)6-s + (0.893 − 0.448i)7-s + (−0.866 − 0.5i)8-s + (−0.835 + 0.549i)9-s + (0.286 + 0.957i)10-s + (−0.286 + 0.957i)11-s + (−0.835 − 0.549i)12-s + (0.448 + 0.893i)13-s + (0.727 + 0.686i)14-s + (0.230 + 0.973i)15-s + (0.173 − 0.984i)16-s + (0.984 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.391 - 0.920i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2917, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.391 - 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-2.100427586 + 3.177212271i\)
\(L(\frac12)\) \(\approx\) \(-2.100427586 + 3.177212271i\)
\(L(1)\) \(\approx\) \(0.7624986028 + 1.469218886i\)
\(L(1)\) \(\approx\) \(0.7624986028 + 1.469218886i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.342 + 0.939i)T \)
3 \( 1 + (0.286 + 0.957i)T \)
5 \( 1 + (0.998 + 0.0581i)T \)
7 \( 1 + (0.893 - 0.448i)T \)
11 \( 1 + (-0.286 + 0.957i)T \)
13 \( 1 + (0.448 + 0.893i)T \)
17 \( 1 + (0.984 + 0.173i)T \)
19 \( 1 + (-0.642 + 0.766i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (-0.230 + 0.973i)T \)
31 \( 1 + (-0.448 + 0.893i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (0.993 + 0.116i)T \)
53 \( 1 + (0.0581 - 0.998i)T \)
59 \( 1 + (0.230 + 0.973i)T \)
61 \( 1 + (-0.116 - 0.993i)T \)
67 \( 1 + (0.893 + 0.448i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (0.286 - 0.957i)T \)
79 \( 1 + (-0.549 - 0.835i)T \)
83 \( 1 + (-0.686 + 0.727i)T \)
89 \( 1 + (-0.802 + 0.597i)T \)
97 \( 1 + (-0.549 + 0.835i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.177951614337678053804124731633, −17.27172283407463624045994167826, −16.97838601117845962459488455641, −15.430443507775418107626527991695, −14.82934960921200725792125539721, −14.10907981942517002884236188015, −13.67325902086569475381720471295, −12.95554839994431998860971722955, −12.59507250148219667057933110572, −11.60669846347432153864948434424, −11.096140334466743760262513538100, −10.41656774004794406798778490862, −9.522242429865238246019828709015, −8.60623657530932705317525449285, −8.45729232337267417294324298894, −7.40719455738786733840906914359, −6.21815363951116991339293960360, −5.69635105265936698588622819788, −5.26675515610319044091091037400, −4.15310649807577444885296914676, −2.9665309604753406752945802487, −2.637011833312725571179098920734, −1.84341970750173656977679779339, −1.00600856746962140624267595876, −0.47514448715743468174968379114, 1.31184968835165718712691002115, 2.10405143507919288841077067425, 3.26717714635456629689072633787, 3.950232831258542106243010602863, 4.739626239195349332387190476, 5.26225886670810192185051833500, 5.86662755087332062108476605773, 6.8924090769193129708221704830, 7.51063900475453450561002584088, 8.401272175949428118516585801263, 8.98366408373515825704660496598, 9.67135254941764039617192268877, 10.34017508052439795976519276621, 10.9953328432713829983040685353, 12.04479915754921609409940611132, 12.83859115450204477524148693328, 13.65403329642460476967328810131, 14.29386812925825240863571262416, 14.57201075929452734047375426943, 15.20100226898451075274014136183, 16.11884425738596450288876429503, 16.72885043688017509101686946734, 17.17172018907916228119400942070, 17.82904488181155467851218018103, 18.47786800555368665643694828292

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