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Properties

Label 2-1120-56.19-c1-0-0
Degree $2$
Conductor $1120$
Sign $-0.859 + 0.510i$
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  + (−1.75 + 1.01i)3-s + (0.5 − 0.866i)5-s + (−1.63 + 2.07i)7-s + (0.552 − 0.957i)9-s + (0.572 + 0.991i)11-s + 0.714·13-s + 2.02i·15-s + (−1.98 + 1.14i)17-s + (3.36 + 1.94i)19-s + (0.770 − 5.30i)21-s + (−4.00 − 2.31i)23-s + (−0.499 − 0.866i)25-s − 3.83i·27-s + 1.54i·29-s + (−0.590 − 1.02i)31-s + ⋯
L(s)  = 1  + (−1.01 + 0.584i)3-s + (0.223 − 0.387i)5-s + (−0.619 + 0.785i)7-s + (0.184 − 0.319i)9-s + (0.172 + 0.299i)11-s + 0.198·13-s + 0.523i·15-s + (−0.480 + 0.277i)17-s + (0.772 + 0.445i)19-s + (0.168 − 1.15i)21-s + (−0.834 − 0.482i)23-s + (−0.0999 − 0.173i)25-s − 0.738i·27-s + 0.286i·29-s + (−0.106 − 0.183i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-0.859 + 0.510i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ -0.859 + 0.510i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1297598428\)
\(L(\frac12)\) \(\approx\) \(0.1297598428\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.63 - 2.07i)T \)
good3 \( 1 + (1.75 - 1.01i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-0.572 - 0.991i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.714T + 13T^{2} \)
17 \( 1 + (1.98 - 1.14i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.36 - 1.94i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.00 + 2.31i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.54iT - 29T^{2} \)
31 \( 1 + (0.590 + 1.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.72 + 3.30i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.9iT - 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + (-2.60 + 4.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.25 - 1.30i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (10.5 - 6.07i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.13 + 12.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.08 + 7.08i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.8iT - 71T^{2} \)
73 \( 1 + (4.71 - 2.72i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.20 + 4.73i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.78iT - 83T^{2} \)
89 \( 1 + (11.0 + 6.38i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.18iT - 97T^{2} \)
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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

−10.21118006239259651149444136155, −9.673705523423134800078986162888, −8.825689997247777173048946040956, −7.957251214317154390140343417147, −6.62231531331044624260942577776, −6.00613716103625174770024528843, −5.25209460499434145063388428476, −4.46722797293780127328234718494, −3.28486183065197608378884471084, −1.85399655200597657973378241448, 0.06615711455099884733520473235, 1.40408318704116869715492189587, 3.01034215714304259288751932593, 4.02199029423174345171244188090, 5.32273609703332189166034854051, 6.03883069338857645567183156185, 6.90440532964911472830072514408, 7.23629878127422993862089242689, 8.513716413890595076741911389930, 9.532631358108921030771736158532

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