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Properties

Label 2-1120-280.13-c1-0-9
Degree $2$
Conductor $1120$
Sign $0.117 + 0.993i$
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  + (−2.42 − 2.42i)3-s + (−1.75 + 1.39i)5-s + (−1.87 + 1.87i)7-s + 8.74i·9-s + (−2.42 − 2.42i)13-s + (7.61 + 0.870i)15-s + 4.21i·19-s + 9.06·21-s + (−0.741 − 0.741i)23-s + (1.12 − 4.87i)25-s + (13.9 − 13.9i)27-s + (0.672 − 5.87i)35-s + 11.7i·39-s + (−12.1 − 15.3i)45-s − 7i·49-s + ⋯
L(s)  = 1  + (−1.39 − 1.39i)3-s + (−0.782 + 0.622i)5-s + (−0.707 + 0.707i)7-s + 2.91i·9-s + (−0.672 − 0.672i)13-s + (1.96 + 0.224i)15-s + 0.968i·19-s + 1.97·21-s + (−0.154 − 0.154i)23-s + (0.225 − 0.974i)25-s + (2.67 − 2.67i)27-s + (0.113 − 0.993i)35-s + 1.88i·39-s + (−1.81 − 2.28i)45-s − i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.117 + 0.993i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ 0.117 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4267763216\)
\(L(\frac12)\) \(\approx\) \(0.4267763216\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.75 - 1.39i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good3 \( 1 + (2.42 + 2.42i)T + 3iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (2.42 + 2.42i)T + 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 - 4.21iT - 19T^{2} \)
23 \( 1 + (0.741 + 0.741i)T + 23iT^{2} \)
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 \( 1 + 0.625iT - 59T^{2} \)
61 \( 1 + 5.47T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 7.22T + 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 15.7iT - 79T^{2} \)
83 \( 1 + (11.4 + 11.4i)T + 83iT^{2} \)
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

−9.987886666724105351404331793650, −8.471323946602347292371080608709, −7.69100808704712742232158245101, −7.11305110289354968744829009505, −6.24585303000526105702642325142, −5.73096115740520698497274537683, −4.69087154863428013564877928139, −3.13190829723628400716069228304, −2.03649869054313773445283064931, −0.37423128158947216270496825623, 0.70559200692970048444893506350, 3.29576215087412504629099815089, 4.19875579638640161630627491565, 4.67729793124261213896263995332, 5.56358761249853124635414488152, 6.60990459229695602744837743932, 7.26545739921210890608633412664, 8.697251085198788281539385608137, 9.515225184868388486138617660848, 9.942390721013077923152190164596

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