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Properties

Label 2-1120-35.9-c1-0-47
Degree $2$
Conductor $1120$
Sign $-0.754 + 0.655i$
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  + (2.04 − 1.17i)3-s + (−0.800 − 2.08i)5-s + (−0.745 − 2.53i)7-s + (1.27 − 2.21i)9-s + (−0.300 − 0.519i)11-s − 3.96i·13-s + (−4.09 − 3.31i)15-s + (−2.53 + 1.46i)17-s + (−1.65 + 2.87i)19-s + (−4.51 − 4.30i)21-s + (−0.370 − 0.213i)23-s + (−3.71 + 3.34i)25-s + 1.04i·27-s + 2.15·29-s + (0.664 + 1.15i)31-s + ⋯
L(s)  = 1  + (1.17 − 0.680i)3-s + (−0.358 − 0.933i)5-s + (−0.281 − 0.959i)7-s + (0.426 − 0.738i)9-s + (−0.0905 − 0.156i)11-s − 1.10i·13-s + (−1.05 − 0.856i)15-s + (−0.614 + 0.354i)17-s + (−0.380 + 0.658i)19-s + (−0.985 − 0.939i)21-s + (−0.0772 − 0.0446i)23-s + (−0.743 + 0.668i)25-s + 0.201i·27-s + 0.399·29-s + (0.119 + 0.206i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.754 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-0.754 + 0.655i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ -0.754 + 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.850955505\)
\(L(\frac12)\) \(\approx\) \(1.850955505\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.800 + 2.08i)T \)
7 \( 1 + (0.745 + 2.53i)T \)
good3 \( 1 + (-2.04 + 1.17i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.300 + 0.519i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.96iT - 13T^{2} \)
17 \( 1 + (2.53 - 1.46i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.65 - 2.87i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.370 + 0.213i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.15T + 29T^{2} \)
31 \( 1 + (-0.664 - 1.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.33 - 3.07i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.99T + 41T^{2} \)
43 \( 1 + 7.79iT - 43T^{2} \)
47 \( 1 + (9.37 + 5.41i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.5 + 6.09i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.98 + 6.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.54 - 6.14i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.7 + 6.19i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.86T + 71T^{2} \)
73 \( 1 + (-14.0 + 8.09i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.76 + 11.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.18iT - 83T^{2} \)
89 \( 1 + (0.395 - 0.684i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.80iT - 97T^{2} \)
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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

−9.339799915247675102214402450664, −8.371546987382163823699613115770, −8.089413981089337750316064371912, −7.30714635602379180189097314733, −6.32374742021588441043446361780, −5.10362600689747143111775248480, −4.00605238809275294941103773693, −3.24205617039870432108326231109, −1.94706241098324284654857873428, −0.68356307038987034093913694289, 2.35418222718476152960460028257, 2.77734908901335176966349801989, 3.91727963617683199091908336938, 4.64876379306705444458793431816, 6.11093555908093293484332485407, 6.83779629847419856971208167233, 7.85522455211030507552121025914, 8.646290571873794814581298962797, 9.384602175931417432585666932555, 9.793961201375122827770675316012

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