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Properties

Label 2-1120-35.4-c1-0-33
Degree $2$
Conductor $1120$
Sign $0.720 - 0.693i$
Analytic cond. $8.94324$
Root an. cond. $2.99052$
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  + (2.73 + 1.58i)3-s + (2.12 − 0.696i)5-s + (2.30 − 1.29i)7-s + (3.49 + 6.06i)9-s + (−2.28 + 3.95i)11-s − 0.903i·13-s + (6.91 + 1.45i)15-s + (−2.49 − 1.44i)17-s + (−2.04 − 3.54i)19-s + (8.36 + 0.0836i)21-s + (−3.63 + 2.09i)23-s + (4.03 − 2.95i)25-s + 12.6i·27-s − 0.822·29-s + (−2.80 + 4.84i)31-s + ⋯
L(s)  = 1  + (1.58 + 0.912i)3-s + (0.950 − 0.311i)5-s + (0.870 − 0.491i)7-s + (1.16 + 2.02i)9-s + (−0.688 + 1.19i)11-s − 0.250i·13-s + (1.78 + 0.375i)15-s + (−0.605 − 0.349i)17-s + (−0.469 − 0.813i)19-s + (1.82 + 0.0182i)21-s + (−0.757 + 0.437i)23-s + (0.806 − 0.591i)25-s + 2.43i·27-s − 0.152·29-s + (−0.502 + 0.871i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.720 - 0.693i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (1089, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ 0.720 - 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.373525185\)
\(L(\frac12)\) \(\approx\) \(3.373525185\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.12 + 0.696i)T \)
7 \( 1 + (-2.30 + 1.29i)T \)
good3 \( 1 + (-2.73 - 1.58i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (2.28 - 3.95i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.903iT - 13T^{2} \)
17 \( 1 + (2.49 + 1.44i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.04 + 3.54i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.63 - 2.09i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.822T + 29T^{2} \)
31 \( 1 + (2.80 - 4.84i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.84 + 5.10i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.31T + 41T^{2} \)
43 \( 1 + 3.07iT - 43T^{2} \)
47 \( 1 + (-5.21 + 3.01i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.24 + 4.75i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.99 + 6.91i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.84 + 8.39i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.62 - 0.936i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.23T + 71T^{2} \)
73 \( 1 + (4.77 + 2.75i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.36 - 5.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.1iT - 83T^{2} \)
89 \( 1 + (3.75 + 6.50i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 17.6iT - 97T^{2} \)
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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

−9.787044771306960351935382064495, −9.211468054776490399219578999777, −8.398737570962795123570807450209, −7.73404435386601055825996198080, −6.85152530085170159600619612317, −5.16100757573877053594422168565, −4.75088060149969299461069487134, −3.78885490409395236210806161438, −2.44279762643274468097774418439, −1.90087192576547467693358936649, 1.50997300337860827596958988227, 2.30238710790167825120584194447, 3.04389849245266681989606833124, 4.30361340155040060034813956283, 5.80165816922689404516448218861, 6.34891804860628505465582089996, 7.52174289675477132051510511625, 8.217131502398610098628012372753, 8.682460447158694195367685655245, 9.474641181599201736711860082941

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