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Properties

Label 2-1120-280.237-c1-0-43
Degree $2$
Conductor $1120$
Sign $-0.904 + 0.425i$
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  + (1.96 − 1.96i)3-s + (0.254 − 2.22i)5-s + (−1.87 − 1.87i)7-s − 4.74i·9-s + (−4.88 + 4.88i)13-s + (−3.87 − 4.87i)15-s − 2.41i·19-s − 7.36·21-s + (6.74 − 6.74i)23-s + (−4.87 − 1.12i)25-s + (−3.42 − 3.42i)27-s + (−4.63 + 3.68i)35-s + 19.2i·39-s + (−10.5 − 1.20i)45-s + 7i·49-s + ⋯
L(s)  = 1  + (1.13 − 1.13i)3-s + (0.113 − 0.993i)5-s + (−0.707 − 0.707i)7-s − 1.58i·9-s + (−1.35 + 1.35i)13-s + (−0.999 − 1.25i)15-s − 0.552i·19-s − 1.60·21-s + (1.40 − 1.40i)23-s + (−0.974 − 0.225i)25-s + (−0.659 − 0.659i)27-s + (−0.782 + 0.622i)35-s + 3.07i·39-s + (−1.57 − 0.179i)45-s + i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-0.904 + 0.425i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ -0.904 + 0.425i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.866245961\)
\(L(\frac12)\) \(\approx\) \(1.866245961\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.254 + 2.22i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good3 \( 1 + (-1.96 + 1.96i)T - 3iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (4.88 - 4.88i)T - 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 + 2.41iT - 19T^{2} \)
23 \( 1 + (-6.74 + 6.74i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + 10.4iT - 59T^{2} \)
61 \( 1 - 6.47T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 15.2T + 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 8.25iT - 79T^{2} \)
83 \( 1 + (12.2 - 12.2i)T - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97iT^{2} \)
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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

−9.274661241747621544947809741975, −8.706525653978998145264451821281, −7.82607386141163618640588624766, −6.94127369455650325828688386147, −6.62935354344275076644179636116, −5.01306648387652536023299991255, −4.15559894726787507964281741132, −2.87636288762870517391529274895, −1.96734492080600686819250883195, −0.68279097708874942884705672610, 2.41458152777243426972978050309, 3.04581784440595423325050525473, 3.65599229508767578457780372132, 5.03231879601749117464921558050, 5.77523916079108207849619256728, 7.07470428673175184516418723889, 7.77800974073797881833693891943, 8.724022066294817996449206203844, 9.601051815325812007330223117806, 9.932845137888075074756215341474

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