↑

Properties

Label 1-1020-1020.1007-r1-0-0
Degree $1$
Conductor $1020$
Sign $0.937 - 0.346i$
Analytic cond. $109.614$
Root an. cond. $109.614$
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  + 7-s + i·11-s − i·13-s − 19-s + 23-s − i·29-s + i·31-s + 37-s − i·41-s − i·43-s + i·47-s + 49-s − i·53-s + 59-s + i·61-s + ⋯
L(s)  = 1  + 7-s + i·11-s − i·13-s − 19-s + 23-s − i·29-s + i·31-s + 37-s − i·41-s − i·43-s + i·47-s + 49-s − i·53-s + 59-s + i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.937 - 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.937 - 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.937 - 0.346i$
Analytic conductor: \(109.614\)
Root analytic conductor: \(109.614\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1020} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1020,\ (1:\ ),\ 0.937 - 0.346i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.401838157 - 0.4300093812i\)
\(L(\frac12)\) \(\approx\) \(2.401838157 - 0.4300093812i\)
\(L(1)\) \(\approx\) \(1.243596501 - 0.04050903117i\)
\(L(1)\) \(\approx\) \(1.243596501 - 0.04050903117i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
19 \( 1 + T \)
23 \( 1 \)
29 \( 1 \)
31 \( 1 \)
37 \( 1 + iT \)
41 \( 1 \)
43 \( 1 - iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 \)
67 \( 1 \)
71 \( 1 - T \)
73 \( 1 \)
79 \( 1 \)
83 \( 1 \)
89 \( 1 + T \)
97 \( 1 \)
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

−21.564595924207365870329047295380, −20.799701087381542573936695341866, −19.92098521393753846380185006135, −18.93103491206463671928568327340, −18.524883247231907052474183580290, −17.45546099992024669738518608203, −16.76964009973404531811172882026, −16.119037769136766413168687912580, −14.857231737759181842312870644843, −14.55045755274230389257999215170, −13.53258572234594174434000489686, −12.83691529209321990398348802395, −11.541186854277688015805688587097, −11.29097061892776858526251090587, −10.35328491482922695657133982934, −9.148546995310465729707632408601, −8.56287705753642100281012664180, −7.704042825067255333165681564349, −6.70153433448390430936858129219, −5.83662877624228762097370175283, −4.80839971011998117591018280288, −4.08748117388278184447876328721, −2.89156919308596179534264676658, −1.83576628091744590449221964619, −0.83851140030608451538261242778, 0.656854326214344873609933291305, 1.81095157201948800682794871000, 2.68258116519757576041417560291, 3.98621561400878084182195245421, 4.81034894992068596502858645351, 5.56517894365207004789925743020, 6.729863449477612269051058842580, 7.59733871930447588191903206037, 8.317109995251592920600040933653, 9.20495370775068627971595520926, 10.29591237462059365426422334648, 10.84774091250862767874552462023, 11.834920626388016934407679110429, 12.6247959749836261900337178267, 13.36246982873655537823210421108, 14.46466518846911655931945391904, 15.028960144230533249192505468652, 15.64557597769380464501194937545, 16.902721255559957725067920224780, 17.52916216152038982343425575964, 18.03839371522105825448338697793, 19.06608110335960054394684578811, 19.85914943287757605737691818950, 20.783277231156725187702965799356, 21.09109894375827589117523967546

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