↑

Properties

Label 1-3301-3301.1056-r1-0-0
Degree $1$
Conductor $3301$
Sign $-0.0865 + 0.996i$
Analytic cond. $354.741$
Root an. cond. $354.741$
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  + (−0.901 − 0.432i)2-s + (0.283 − 0.958i)3-s + (0.625 + 0.780i)4-s + (0.832 − 0.553i)5-s + (−0.670 + 0.741i)6-s + (0.0741 − 0.997i)7-s + (−0.226 − 0.974i)8-s + (−0.839 − 0.543i)9-s + (−0.990 + 0.138i)10-s + (−0.00951 − 0.999i)11-s + (0.925 − 0.378i)12-s + (0.411 − 0.911i)13-s + (−0.498 + 0.866i)14-s + (−0.294 − 0.955i)15-s + (−0.217 + 0.976i)16-s + (0.385 − 0.922i)17-s + ⋯
L(s)  = 1  + (−0.901 − 0.432i)2-s + (0.283 − 0.958i)3-s + (0.625 + 0.780i)4-s + (0.832 − 0.553i)5-s + (−0.670 + 0.741i)6-s + (0.0741 − 0.997i)7-s + (−0.226 − 0.974i)8-s + (−0.839 − 0.543i)9-s + (−0.990 + 0.138i)10-s + (−0.00951 − 0.999i)11-s + (0.925 − 0.378i)12-s + (0.411 − 0.911i)13-s + (−0.498 + 0.866i)14-s + (−0.294 − 0.955i)15-s + (−0.217 + 0.976i)16-s + (0.385 − 0.922i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3301 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0865 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3301 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0865 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3301\)
Sign: $-0.0865 + 0.996i$
Analytic conductor: \(354.741\)
Root analytic conductor: \(354.741\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3301} (1056, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3301,\ (1:\ ),\ -0.0865 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.304453363 - 1.422698847i\)
\(L(\frac12)\) \(\approx\) \(-1.304453363 - 1.422698847i\)
\(L(1)\) \(\approx\) \(0.4902910812 - 0.8236204056i\)
\(L(1)\) \(\approx\) \(0.4902910812 - 0.8236204056i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3301 \( 1 \)
good2 \( 1 + (-0.901 - 0.432i)T \)
3 \( 1 + (0.283 - 0.958i)T \)
5 \( 1 + (0.832 - 0.553i)T \)
7 \( 1 + (0.0741 - 0.997i)T \)
11 \( 1 + (-0.00951 - 0.999i)T \)
13 \( 1 + (0.411 - 0.911i)T \)
17 \( 1 + (0.385 - 0.922i)T \)
19 \( 1 + (0.915 - 0.401i)T \)
23 \( 1 + (0.694 - 0.719i)T \)
29 \( 1 + (-0.788 - 0.615i)T \)
31 \( 1 + (-0.991 + 0.130i)T \)
37 \( 1 + (-0.974 - 0.224i)T \)
41 \( 1 + (0.666 - 0.745i)T \)
43 \( 1 + (0.979 - 0.202i)T \)
47 \( 1 + (0.998 + 0.0551i)T \)
53 \( 1 + (0.576 + 0.816i)T \)
59 \( 1 + (-0.427 + 0.904i)T \)
61 \( 1 + (-0.328 + 0.944i)T \)
67 \( 1 + (0.641 + 0.766i)T \)
71 \( 1 + (-0.155 - 0.987i)T \)
73 \( 1 + (-0.228 + 0.973i)T \)
79 \( 1 + (-0.401 - 0.915i)T \)
83 \( 1 + (-0.391 + 0.920i)T \)
89 \( 1 + (-0.439 - 0.898i)T \)
97 \( 1 + (0.849 - 0.527i)T \)
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

−18.947174441518392308118183717452, −18.54835546925916046931055950130, −17.73645080301550078053156147814, −17.16097848551804710967628223842, −16.47821337328147101297295168507, −15.69576089038920854852029976545, −15.17093653657187784032804539708, −14.47967221900843232987607181264, −14.18499268661717735279066553930, −12.987893543948028455601909434541, −11.95418969247449933198995057123, −11.13779837565269006641853186236, −10.63909061263001507443107141508, −9.70040009595083967815730706001, −9.39557073284602744230248997760, −8.88340946210132165480154246802, −7.91117368687265453236052912781, −7.13722461010973660695285567017, −6.25016586885248983661593090566, −5.51248822867415767940184410526, −5.10405869880295282658897318654, −3.78442944918270927108464860235, −2.89800628577489099354335702832, −1.97068190676783840588967061299, −1.54590340398638524744741357657, 0.45620916536322347222547547452, 0.79656982662829189142625118671, 1.44359001643195028079493530729, 2.5574643779674023733736367728, 3.09625689490304718706571080426, 4.029840828141802175415376673967, 5.47700026203457098515669190810, 5.95258593497964497052108838027, 7.19498499030964939476156942034, 7.342567841648060432756619137399, 8.338758701483200649205940579338, 8.95309612762822204679911966941, 9.49492120296565212463739849375, 10.55816557470600226474277387403, 10.95468823450162898808745351726, 11.89184942357950916444514756837, 12.61699823968874143970943514718, 13.33305058876689119450074274349, 13.69876792285492959173796612924, 14.44171437331451928009026664017, 15.71372014333215044986003644891, 16.43027249750664291180509407593, 17.05451294494910139225917450185, 17.55960964498409477535796505025, 18.26738627898403884083704757376

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