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Properties

Label 2-1120-56.3-c1-0-0
Degree $2$
Conductor $1120$
Sign $0.479 - 0.877i$
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  + (−2.75 − 1.58i)3-s + (−0.5 − 0.866i)5-s + (1.04 − 2.43i)7-s + (3.55 + 6.15i)9-s + (−1.21 + 2.09i)11-s − 1.53·13-s + 3.17i·15-s + (−6.58 − 3.79i)17-s + (1.52 − 0.883i)19-s + (−6.73 + 5.03i)21-s + (−5.66 + 3.26i)23-s + (−0.499 + 0.866i)25-s − 13.0i·27-s + 2.34i·29-s + (1.04 − 1.80i)31-s + ⋯
L(s)  = 1  + (−1.58 − 0.917i)3-s + (−0.223 − 0.387i)5-s + (0.394 − 0.919i)7-s + (1.18 + 2.05i)9-s + (−0.364 + 0.631i)11-s − 0.426·13-s + 0.820i·15-s + (−1.59 − 0.921i)17-s + (0.350 − 0.202i)19-s + (−1.47 + 1.09i)21-s + (−1.18 + 0.681i)23-s + (−0.0999 + 0.173i)25-s − 2.51i·27-s + 0.435i·29-s + (0.187 − 0.323i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.479 - 0.877i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ 0.479 - 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2603900126\)
\(L(\frac12)\) \(\approx\) \(0.2603900126\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1.04 + 2.43i)T \)
good3 \( 1 + (2.75 + 1.58i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.21 - 2.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.53T + 13T^{2} \)
17 \( 1 + (6.58 + 3.79i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.52 + 0.883i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.66 - 3.26i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.34iT - 29T^{2} \)
31 \( 1 + (-1.04 + 1.80i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.47 + 1.42i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.90iT - 41T^{2} \)
43 \( 1 - 1.39T + 43T^{2} \)
47 \( 1 + (-5.65 - 9.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.82 - 3.93i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.90 + 2.82i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.93 - 8.54i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.13 - 5.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.98iT - 71T^{2} \)
73 \( 1 + (2.73 + 1.57i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.75 + 1.01i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.288iT - 83T^{2} \)
89 \( 1 + (-12.0 + 6.95i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.249iT - 97T^{2} \)
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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

−10.24870148615754008411117912903, −9.269524914819579396798885151674, −7.84351196545222228271073254529, −7.41029410174562213802525852603, −6.71579235733352560070291393020, −5.76252486935395651328956648872, −4.79184980747345330776160142476, −4.35254054428131006819398139954, −2.25621093558857845365237097556, −1.04607299538169291990461977907, 0.16646414660236373848071773828, 2.24954283835885874159371564377, 3.79228946200153616058330079120, 4.58702797904977884006149167671, 5.49587348452178764456209322587, 6.09079458228482750960601836972, 6.84478119714615943606922682298, 8.178201060040001865382059809723, 8.978073853179799674061548052355, 9.982811987394112556353363523635

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