↑

Properties

Label 2-1122-17.13-c1-0-26
Degree $2$
Conductor $1122$
Sign $-0.788 + 0.615i$
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 + (−2.41 − 2.41i)5-s + (0.707 − 0.707i)6-s + (1.41 − 1.41i)7-s + i·8-s + 1.00i·9-s + (−2.41 + 2.41i)10-s + (−0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s + 6.82·13-s + (−1.41 − 1.41i)14-s − 3.41i·15-s + 16-s + (1 − 4i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (−1.07 − 1.07i)5-s + (0.288 − 0.288i)6-s + (0.534 − 0.534i)7-s + 0.353i·8-s + 0.333i·9-s + (−0.763 + 0.763i)10-s + (−0.213 + 0.213i)11-s + (−0.204 − 0.204i)12-s + 1.89·13-s + (−0.377 − 0.377i)14-s − 0.881i·15-s + 0.250·16-s + (0.242 − 0.970i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1122\)    =    \(2 \cdot 3 \cdot 11 \cdot 17\)
Sign: $-0.788 + 0.615i$
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: \((2,\ 1122,\ (\ :1/2),\ -0.788 + 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.299067952\)
\(L(\frac12)\) \(\approx\) \(1.299067952\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 4i)T \)
good5 \( 1 + (2.41 + 2.41i)T + 5iT^{2} \)
7 \( 1 + (-1.41 + 1.41i)T - 7iT^{2} \)
13 \( 1 - 6.82T + 13T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (1.41 - 1.41i)T - 23iT^{2} \)
29 \( 1 + (6.41 + 6.41i)T + 29iT^{2} \)
31 \( 1 + (5.41 + 5.41i)T + 31iT^{2} \)
37 \( 1 + (6.41 + 6.41i)T + 37iT^{2} \)
41 \( 1 + (-3 + 3i)T - 41iT^{2} \)
43 \( 1 + 6.82iT - 43T^{2} \)
47 \( 1 + 9.65T + 47T^{2} \)
53 \( 1 - 12.4iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (-0.757 + 0.757i)T - 61iT^{2} \)
67 \( 1 - 8.48T + 67T^{2} \)
71 \( 1 + (11.0 + 11.0i)T + 71iT^{2} \)
73 \( 1 + (-4.65 - 4.65i)T + 73iT^{2} \)
79 \( 1 + (-8.24 + 8.24i)T - 79iT^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 - 17.6T + 89T^{2} \)
97 \( 1 + (-7.48 - 7.48i)T + 97iT^{2} \)
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

−9.259358754907019062260129575815, −8.921611863034158779195120914467, −7.897965759506762929797284841447, −7.56394479071440954431478954308, −5.79487652993294290449529820411, −4.85043047189378586843553359091, −3.96527847944736682507779426819, −3.59087826943995828849401372559, −1.86187525443643589140165478519, −0.57706765422531386582597211758, 1.62798071945213323493123660608, 3.36446882913821603442399330151, 3.69393740253842071742392216317, 5.16996779989628302553150782320, 6.23847485991616313476476457083, 6.82040363777125489535672878845, 7.78587441945212453720437085480, 8.363111735644091480313214207320, 8.819863183975932815744480093597, 10.21733583244204197801842212739

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