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Properties

Label 2-1122-17.13-c1-0-3
Degree $2$
Conductor $1122$
Sign $-0.383 - 0.923i$
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 + (2.24 + 2.24i)5-s + (0.707 − 0.707i)6-s + (−2.78 + 2.78i)7-s + i·8-s + 1.00i·9-s + (2.24 − 2.24i)10-s + (0.707 − 0.707i)11-s + (−0.707 − 0.707i)12-s − 5.93·13-s + (2.78 + 2.78i)14-s + 3.17i·15-s + 16-s + (−3.59 − 2.02i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (1.00 + 1.00i)5-s + (0.288 − 0.288i)6-s + (−1.05 + 1.05i)7-s + 0.353i·8-s + 0.333i·9-s + (0.708 − 0.708i)10-s + (0.213 − 0.213i)11-s + (−0.204 − 0.204i)12-s − 1.64·13-s + (0.744 + 0.744i)14-s + 0.818i·15-s + 0.250·16-s + (−0.870 − 0.491i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1122\)    =    \(2 \cdot 3 \cdot 11 \cdot 17\)
Sign: $-0.383 - 0.923i$
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: \((2,\ 1122,\ (\ :1/2),\ -0.383 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.166165016\)
\(L(\frac12)\) \(\approx\) \(1.166165016\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.59 + 2.02i)T \)
good5 \( 1 + (-2.24 - 2.24i)T + 5iT^{2} \)
7 \( 1 + (2.78 - 2.78i)T - 7iT^{2} \)
13 \( 1 + 5.93T + 13T^{2} \)
19 \( 1 - 6.44iT - 19T^{2} \)
23 \( 1 + (-3.36 + 3.36i)T - 23iT^{2} \)
29 \( 1 + (-0.809 - 0.809i)T + 29iT^{2} \)
31 \( 1 + (4.37 + 4.37i)T + 31iT^{2} \)
37 \( 1 + (1.67 + 1.67i)T + 37iT^{2} \)
41 \( 1 + (-3.98 + 3.98i)T - 41iT^{2} \)
43 \( 1 - 5.64iT - 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 - 5.77iT - 53T^{2} \)
59 \( 1 - 9.05iT - 59T^{2} \)
61 \( 1 + (5.48 - 5.48i)T - 61iT^{2} \)
67 \( 1 - 3.61T + 67T^{2} \)
71 \( 1 + (-11.1 - 11.1i)T + 71iT^{2} \)
73 \( 1 + (-0.569 - 0.569i)T + 73iT^{2} \)
79 \( 1 + (-11.7 + 11.7i)T - 79iT^{2} \)
83 \( 1 - 2.28iT - 83T^{2} \)
89 \( 1 - 9.65T + 89T^{2} \)
97 \( 1 + (-8.73 - 8.73i)T + 97iT^{2} \)
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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

−10.04762253670572827463047730269, −9.385947878765019875978541622095, −8.987363866126936985665967626611, −7.64761800695405715165147063579, −6.58789167870667675053450009123, −5.85803581068997053257266573817, −4.88900705630744933097167448378, −3.55330598928060952086951389954, −2.60721786510298841316666838589, −2.24588259133591508460252232148, 0.44651188519164248484920409850, 1.95202126047662963579418979080, 3.32040055289515531219221573354, 4.64070166630260833387470591381, 5.20518446708259946018887062900, 6.59925998676631630495277178173, 6.85143163926193182260664509014, 7.79659908199375876723661455289, 8.913303909100712429193068291092, 9.471717063729357951294095959826

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