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Properties

Label 2-1120-140.139-c1-0-22
Degree $2$
Conductor $1120$
Sign $0.448 - 0.893i$
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  + 1.86i·3-s + (−2.07 + 0.836i)5-s + (2.64 + 0.168i)7-s − 0.486·9-s − 4.26i·11-s + 5.57·13-s + (−1.56 − 3.87i)15-s + 3.01·17-s + 0.0464·19-s + (−0.314 + 4.93i)21-s + 5.58·23-s + (3.60 − 3.46i)25-s + 4.69i·27-s + 1.21·29-s − 4.62·31-s + ⋯
L(s)  = 1  + 1.07i·3-s + (−0.927 + 0.374i)5-s + (0.997 + 0.0635i)7-s − 0.162·9-s − 1.28i·11-s + 1.54·13-s + (−0.403 − 0.999i)15-s + 0.731·17-s + 0.0106·19-s + (−0.0685 + 1.07i)21-s + 1.16·23-s + (0.720 − 0.693i)25-s + 0.903i·27-s + 0.224·29-s − 0.829·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.448 - 0.893i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (1119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ 0.448 - 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.747709373\)
\(L(\frac12)\) \(\approx\) \(1.747709373\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.07 - 0.836i)T \)
7 \( 1 + (-2.64 - 0.168i)T \)
good3 \( 1 - 1.86iT - 3T^{2} \)
11 \( 1 + 4.26iT - 11T^{2} \)
13 \( 1 - 5.57T + 13T^{2} \)
17 \( 1 - 3.01T + 17T^{2} \)
19 \( 1 - 0.0464T + 19T^{2} \)
23 \( 1 - 5.58T + 23T^{2} \)
29 \( 1 - 1.21T + 29T^{2} \)
31 \( 1 + 4.62T + 31T^{2} \)
37 \( 1 - 9.87iT - 37T^{2} \)
41 \( 1 + 10.5iT - 41T^{2} \)
43 \( 1 + 12.4T + 43T^{2} \)
47 \( 1 - 6.78iT - 47T^{2} \)
53 \( 1 + 4.99iT - 53T^{2} \)
59 \( 1 + 7.06T + 59T^{2} \)
61 \( 1 - 11.5iT - 61T^{2} \)
67 \( 1 - 6.65T + 67T^{2} \)
71 \( 1 + 5.11iT - 71T^{2} \)
73 \( 1 - 9.67T + 73T^{2} \)
79 \( 1 - 5.90iT - 79T^{2} \)
83 \( 1 + 11.1iT - 83T^{2} \)
89 \( 1 + 1.51iT - 89T^{2} \)
97 \( 1 - 3.01T + 97T^{2} \)
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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

−10.21804866107439310591959347802, −8.919183495047798373423526475008, −8.512374716463769205147303309541, −7.70383700012978788384978571848, −6.61447844644994936687057905283, −5.50860064386439196540013872650, −4.71495934451799089560764694990, −3.67652694047798490814339994960, −3.24387709354308176981192950925, −1.18557525061130973432887429427, 1.05776199192834687932664329246, 1.85316978272782936827160900765, 3.49545415217020654445558337524, 4.48684095313564009069785223098, 5.33784413124594559360368168967, 6.62655481247090588942240093410, 7.30658630381981923904089277919, 7.967007290239488193087517064852, 8.532962590863966824644561810089, 9.560742369698336636392967599140

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