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Properties

Label 2-1122-17.13-c1-0-18
Degree $2$
Conductor $1122$
Sign $0.956 + 0.293i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + (0.707 + 0.707i)3-s − 4-s + (2.11 + 2.11i)5-s + (0.707 − 0.707i)6-s + (1.83 − 1.83i)7-s + i·8-s + 1.00i·9-s + (2.11 − 2.11i)10-s + (0.707 − 0.707i)11-s + (−0.707 − 0.707i)12-s + 0.600·13-s + (−1.83 − 1.83i)14-s + 2.99i·15-s + 16-s + (3.85 − 1.47i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (0.945 + 0.945i)5-s + (0.288 − 0.288i)6-s + (0.694 − 0.694i)7-s + 0.353i·8-s + 0.333i·9-s + (0.668 − 0.668i)10-s + (0.213 − 0.213i)11-s + (−0.204 − 0.204i)12-s + 0.166·13-s + (−0.491 − 0.491i)14-s + 0.772i·15-s + 0.250·16-s + (0.934 − 0.357i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1122\)    =    \(2 \cdot 3 \cdot 11 \cdot 17\)
Sign: $0.956 + 0.293i$
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: \((2,\ 1122,\ (\ :1/2),\ 0.956 + 0.293i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.341772107\)
\(L(\frac12)\) \(\approx\) \(2.341772107\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.85 + 1.47i)T \)
good5 \( 1 + (-2.11 - 2.11i)T + 5iT^{2} \)
7 \( 1 + (-1.83 + 1.83i)T - 7iT^{2} \)
13 \( 1 - 0.600T + 13T^{2} \)
19 \( 1 + 7.35iT - 19T^{2} \)
23 \( 1 + (4.88 - 4.88i)T - 23iT^{2} \)
29 \( 1 + (-5.63 - 5.63i)T + 29iT^{2} \)
31 \( 1 + (2.31 + 2.31i)T + 31iT^{2} \)
37 \( 1 + (-3.40 - 3.40i)T + 37iT^{2} \)
41 \( 1 + (-6.53 + 6.53i)T - 41iT^{2} \)
43 \( 1 - 9.98iT - 43T^{2} \)
47 \( 1 + 8.13T + 47T^{2} \)
53 \( 1 - 2.55iT - 53T^{2} \)
59 \( 1 + 8.35iT - 59T^{2} \)
61 \( 1 + (4.05 - 4.05i)T - 61iT^{2} \)
67 \( 1 - 13.2T + 67T^{2} \)
71 \( 1 + (0.838 + 0.838i)T + 71iT^{2} \)
73 \( 1 + (10.9 + 10.9i)T + 73iT^{2} \)
79 \( 1 + (3.88 - 3.88i)T - 79iT^{2} \)
83 \( 1 - 15.9iT - 83T^{2} \)
89 \( 1 - 14.1T + 89T^{2} \)
97 \( 1 + (1.72 + 1.72i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.739281998172853831387616430180, −9.404633559662975452188320582553, −8.232849668762766748361588829786, −7.41487062324157366533889400848, −6.42044805344457855554975427483, −5.34113133945593376491133699916, −4.42996994207765509265477800807, −3.32604581298188086260691733056, −2.54987909978784782123061964465, −1.32367062200864762249832095939, 1.29549672830221296454554966520, 2.22709240319270198418104815560, 3.87525538608020220340542249211, 4.91907035253501941414272751869, 5.83812311340114944756494911765, 6.21372470281980419315851658660, 7.62889824531044512627766678986, 8.299722117283737089350165410063, 8.742436937503861875193250840372, 9.744016846619529137593684603267

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