Properties

Label 2-1120-56.3-c1-0-23
Degree $2$
Conductor $1120$
Sign $-0.724 + 0.689i$
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  + (−1.90 − 1.10i)3-s + (−0.5 − 0.866i)5-s + (0.584 − 2.58i)7-s + (0.922 + 1.59i)9-s + (2.90 − 5.03i)11-s + 4.83·13-s + 2.20i·15-s + (3.78 + 2.18i)17-s + (−1.63 + 0.945i)19-s + (−3.95 + 4.27i)21-s + (−0.157 + 0.0911i)23-s + (−0.499 + 0.866i)25-s + 2.54i·27-s − 4.38i·29-s + (2.03 − 3.53i)31-s + ⋯
L(s)  = 1  + (−1.10 − 0.635i)3-s + (−0.223 − 0.387i)5-s + (0.220 − 0.975i)7-s + (0.307 + 0.532i)9-s + (0.875 − 1.51i)11-s + 1.34·13-s + 0.568i·15-s + (0.917 + 0.529i)17-s + (−0.375 + 0.216i)19-s + (−0.862 + 0.933i)21-s + (−0.0329 + 0.0189i)23-s + (−0.0999 + 0.173i)25-s + 0.489i·27-s − 0.815i·29-s + (0.366 − 0.634i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.724 + 0.689i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-0.724 + 0.689i$
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: \((2,\ 1120,\ (\ :1/2),\ -0.724 + 0.689i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.106824200\)
\(L(\frac12)\) \(\approx\) \(1.106824200\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.584 + 2.58i)T \)
good3 \( 1 + (1.90 + 1.10i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-2.90 + 5.03i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.83T + 13T^{2} \)
17 \( 1 + (-3.78 - 2.18i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.63 - 0.945i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.157 - 0.0911i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.38iT - 29T^{2} \)
31 \( 1 + (-2.03 + 3.53i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.69 + 2.13i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.44iT - 41T^{2} \)
43 \( 1 - 2.10T + 43T^{2} \)
47 \( 1 + (0.946 + 1.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.54 + 4.93i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.35 - 3.66i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.89 + 8.47i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.04 - 10.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.80iT - 71T^{2} \)
73 \( 1 + (-7.34 - 4.24i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.5 - 6.09i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.9iT - 83T^{2} \)
89 \( 1 + (5.11 - 2.95i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.2iT - 97T^{2} \)
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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

−9.561697390823285609393616060907, −8.448366097398117384876720074049, −7.949734174022411822421256582152, −6.78637273703269056574650586964, −6.10359695503412923841560297607, −5.57882410539371465415481785634, −4.15448971803787826850405778090, −3.52054503112180027249144104187, −1.34717707803800662820582099602, −0.67655930513769048229966988504, 1.52228180276319748174611331382, 3.04494356723540391206038002060, 4.29109405549648123115414360129, 4.95467318649816391681707239413, 5.94565839290738956215308985625, 6.53285129038338492414167699331, 7.58170487358049834763275298085, 8.656494127576536192622566936597, 9.467304826026949278776399002284, 10.21785852199577364339488566397

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