↑

Properties

Label 2-1122-17.13-c1-0-30
Degree $2$
Conductor $1122$
Sign $-0.945 - 0.325i$
Analytic cond. $8.95921$
Root an. cond. $2.99319$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + (0.707 + 0.707i)3-s − 4-s + (−1.28 − 1.28i)5-s + (0.707 − 0.707i)6-s + (−0.233 + 0.233i)7-s + i·8-s + 1.00i·9-s + (−1.28 + 1.28i)10-s + (0.707 − 0.707i)11-s + (−0.707 − 0.707i)12-s − 2.33·13-s + (0.233 + 0.233i)14-s − 1.82i·15-s + 16-s + (−3.89 + 1.34i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (−0.575 − 0.575i)5-s + (0.288 − 0.288i)6-s + (−0.0884 + 0.0884i)7-s + 0.353i·8-s + 0.333i·9-s + (−0.407 + 0.407i)10-s + (0.213 − 0.213i)11-s + (−0.204 − 0.204i)12-s − 0.646·13-s + (0.0625 + 0.0625i)14-s − 0.470i·15-s + 0.250·16-s + (−0.945 + 0.325i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1122\)    =    \(2 \cdot 3 \cdot 11 \cdot 17\)
Sign: $-0.945 - 0.325i$
Analytic conductor: \(8.95921\)
Root analytic conductor: \(2.99319\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1122} (727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1122,\ (\ :1/2),\ -0.945 - 0.325i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3358107438\)
\(L(\frac12)\) \(\approx\) \(0.3358107438\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (3.89 - 1.34i)T \)
good5 \( 1 + (1.28 + 1.28i)T + 5iT^{2} \)
7 \( 1 + (0.233 - 0.233i)T - 7iT^{2} \)
13 \( 1 + 2.33T + 13T^{2} \)
19 \( 1 + 5.10iT - 19T^{2} \)
23 \( 1 + (3.88 - 3.88i)T - 23iT^{2} \)
29 \( 1 + (2.09 + 2.09i)T + 29iT^{2} \)
31 \( 1 + (5.14 + 5.14i)T + 31iT^{2} \)
37 \( 1 + (-2.47 - 2.47i)T + 37iT^{2} \)
41 \( 1 + (7.09 - 7.09i)T - 41iT^{2} \)
43 \( 1 + 5.67iT - 43T^{2} \)
47 \( 1 + 6.15T + 47T^{2} \)
53 \( 1 + 9.84iT - 53T^{2} \)
59 \( 1 - 13.1iT - 59T^{2} \)
61 \( 1 + (-0.510 + 0.510i)T - 61iT^{2} \)
67 \( 1 + 2.19T + 67T^{2} \)
71 \( 1 + (2.26 + 2.26i)T + 71iT^{2} \)
73 \( 1 + (-0.663 - 0.663i)T + 73iT^{2} \)
79 \( 1 + (6.38 - 6.38i)T - 79iT^{2} \)
83 \( 1 + 5.92iT - 83T^{2} \)
89 \( 1 + 0.734T + 89T^{2} \)
97 \( 1 + (9.12 + 9.12i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.453282454588132644484809071546, −8.681521044017699101035863854297, −8.041242423836581006128692051551, −7.02632242942666936054531955493, −5.74713435357891619441571442220, −4.64929992666096188679116970609, −4.11471523528175588170971938479, −3.02249694560030141367667548911, −1.91279862163347651260619570869, −0.13229715536883502563611082176, 1.92706854667355917775102918485, 3.28555904622441256761212069465, 4.13650289779904516073133587876, 5.24650460809796396847368891624, 6.39424882937647419717491688773, 7.03863173932147836035082516926, 7.67117319107386545517251828705, 8.459252461388959034035985898125, 9.284489944213085200847780183817, 10.14822222664223194420468577650

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