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Properties

Label 2-1120-56.3-c1-0-1
Degree $2$
Conductor $1120$
Sign $-0.975 - 0.217i$
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  + (0.219 + 0.126i)3-s + (0.5 + 0.866i)5-s + (0.978 + 2.45i)7-s + (−1.46 − 2.54i)9-s + (−1.81 + 3.14i)11-s − 5.36·13-s + 0.253i·15-s + (−4.46 − 2.58i)17-s + (−5.49 + 3.17i)19-s + (−0.0966 + 0.663i)21-s + (0.231 − 0.133i)23-s + (−0.499 + 0.866i)25-s − 1.50i·27-s + 2.99i·29-s + (2.72 − 4.71i)31-s + ⋯
L(s)  = 1  + (0.126 + 0.0731i)3-s + (0.223 + 0.387i)5-s + (0.369 + 0.929i)7-s + (−0.489 − 0.847i)9-s + (−0.547 + 0.947i)11-s − 1.48·13-s + 0.0654i·15-s + (−1.08 − 0.625i)17-s + (−1.25 + 0.727i)19-s + (−0.0210 + 0.144i)21-s + (0.0482 − 0.0278i)23-s + (−0.0999 + 0.173i)25-s − 0.289i·27-s + 0.556i·29-s + (0.489 − 0.847i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.217i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-0.975 - 0.217i$
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: \((2,\ 1120,\ (\ :1/2),\ -0.975 - 0.217i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5227076440\)
\(L(\frac12)\) \(\approx\) \(0.5227076440\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.978 - 2.45i)T \)
good3 \( 1 + (-0.219 - 0.126i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.81 - 3.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.36T + 13T^{2} \)
17 \( 1 + (4.46 + 2.58i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.49 - 3.17i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.231 + 0.133i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.99iT - 29T^{2} \)
31 \( 1 + (-2.72 + 4.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.48 + 4.32i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 + 5.33T + 43T^{2} \)
47 \( 1 + (-2.26 - 3.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.09 + 1.78i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.83 + 3.94i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.63 - 4.55i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.963 - 1.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.9iT - 71T^{2} \)
73 \( 1 + (-7.12 - 4.11i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.94 - 5.74i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.75iT - 83T^{2} \)
89 \( 1 + (-4.77 + 2.75i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.98iT - 97T^{2} \)
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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

−10.01232717962929913996301939227, −9.452815699797423882619013714928, −8.672356822396001840658625743579, −7.75064682547307612483146485514, −6.85813271567211896832968117659, −6.02666902333063705661844572187, −5.06089047769702501594191091272, −4.21488607333578896676004053915, −2.65746357900728015319357186444, −2.21063812611030794443024239506, 0.20114311854823161079528333581, 1.96803951938353240413994619103, 2.93279026681041980427439823232, 4.50035350265542601745579389747, 4.87301224093536633692524985208, 6.08828275187317240104205460164, 6.99635614645338165612034538622, 8.033547134891131448732261601908, 8.389532536888608047959082481292, 9.434697216416832484990257621476

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