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Properties

Label 2-1122-17.13-c1-0-29
Degree $2$
Conductor $1122$
Sign $-0.999 - 0.000253i$
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 + (0.680 + 0.680i)5-s + (−0.707 + 0.707i)6-s + (2.13 − 2.13i)7-s + i·8-s + 1.00i·9-s + (0.680 − 0.680i)10-s + (−0.707 + 0.707i)11-s + (0.707 + 0.707i)12-s − 5.01·13-s + (−2.13 − 2.13i)14-s − 0.962i·15-s + 16-s + (−3.25 + 2.53i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (0.304 + 0.304i)5-s + (−0.288 + 0.288i)6-s + (0.805 − 0.805i)7-s + 0.353i·8-s + 0.333i·9-s + (0.215 − 0.215i)10-s + (−0.213 + 0.213i)11-s + (0.204 + 0.204i)12-s − 1.39·13-s + (−0.569 − 0.569i)14-s − 0.248i·15-s + 0.250·16-s + (−0.788 + 0.615i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.000253i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.000253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1122\)    =    \(2 \cdot 3 \cdot 11 \cdot 17\)
Sign: $-0.999 - 0.000253i$
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: \((2,\ 1122,\ (\ :1/2),\ -0.999 - 0.000253i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8701666194\)
\(L(\frac12)\) \(\approx\) \(0.8701666194\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.25 - 2.53i)T \)
good5 \( 1 + (-0.680 - 0.680i)T + 5iT^{2} \)
7 \( 1 + (-2.13 + 2.13i)T - 7iT^{2} \)
13 \( 1 + 5.01T + 13T^{2} \)
19 \( 1 + 4.17iT - 19T^{2} \)
23 \( 1 + (-5.48 + 5.48i)T - 23iT^{2} \)
29 \( 1 + (2.53 + 2.53i)T + 29iT^{2} \)
31 \( 1 + (5.25 + 5.25i)T + 31iT^{2} \)
37 \( 1 + (1.89 + 1.89i)T + 37iT^{2} \)
41 \( 1 + (-3.67 + 3.67i)T - 41iT^{2} \)
43 \( 1 - 6.42iT - 43T^{2} \)
47 \( 1 + 1.45T + 47T^{2} \)
53 \( 1 + 3.72iT - 53T^{2} \)
59 \( 1 - 0.775iT - 59T^{2} \)
61 \( 1 + (8.49 - 8.49i)T - 61iT^{2} \)
67 \( 1 + 3.07T + 67T^{2} \)
71 \( 1 + (8.60 + 8.60i)T + 71iT^{2} \)
73 \( 1 + (9.36 + 9.36i)T + 73iT^{2} \)
79 \( 1 + (0.994 - 0.994i)T - 79iT^{2} \)
83 \( 1 - 7.17iT - 83T^{2} \)
89 \( 1 + 1.45T + 89T^{2} \)
97 \( 1 + (-2.73 - 2.73i)T + 97iT^{2} \)
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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

−9.589804011496998595671477580287, −8.701812424608290967307206640574, −7.61223683120351919585360079780, −7.08878587063380078831508375239, −6.00280097350956300339136365469, −4.77419924775322412043783587433, −4.38833987352838001198752624764, −2.72282270146577827866360116202, −1.91329550155386971564542440204, −0.39064870465434430799237704063, 1.70371656489648048118580799050, 3.18153094680937598985995614339, 4.61910771232787059572985596331, 5.26315326973493466878532066555, 5.68756643523821084801007495263, 7.02083822757607851036263354893, 7.61904043925724561284182900819, 8.802433290180415512487051719482, 9.191047840915334486260803217365, 10.05980343589049830586251291881

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