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Properties

Label 2-560-35.27-c1-0-7
Degree $2$
Conductor $560$
Sign $0.0103 - 0.999i$
Analytic cond. $4.47162$
Root an. cond. $2.11462$
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  + (1.58 + 1.58i)3-s + (1.58 + 1.58i)5-s + (−2.58 + 0.581i)7-s + 2.00i·9-s + 11-s + (1.58 + 1.58i)13-s + 5.00i·15-s + (−1.58 + 1.58i)17-s + 3.16·19-s + (−5 − 3.16i)21-s + (−2 + 2i)23-s + 5.00i·25-s + (1.58 − 1.58i)27-s + 3i·29-s − 3.16i·31-s + ⋯
L(s)  = 1  + (0.912 + 0.912i)3-s + (0.707 + 0.707i)5-s + (−0.975 + 0.219i)7-s + 0.666i·9-s + 0.301·11-s + (0.438 + 0.438i)13-s + 1.29i·15-s + (−0.383 + 0.383i)17-s + 0.725·19-s + (−1.09 − 0.690i)21-s + (−0.417 + 0.417i)23-s + 1.00i·25-s + (0.304 − 0.304i)27-s + 0.557i·29-s − 0.567i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0103 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0103 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.0103 - 0.999i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.0103 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41502 + 1.40042i\)
\(L(\frac12)\) \(\approx\) \(1.41502 + 1.40042i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.58 - 1.58i)T \)
7 \( 1 + (2.58 - 0.581i)T \)
good3 \( 1 + (-1.58 - 1.58i)T + 3iT^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 + (-1.58 - 1.58i)T + 13iT^{2} \)
17 \( 1 + (1.58 - 1.58i)T - 17iT^{2} \)
19 \( 1 - 3.16T + 19T^{2} \)
23 \( 1 + (2 - 2i)T - 23iT^{2} \)
29 \( 1 - 3iT - 29T^{2} \)
31 \( 1 + 3.16iT - 31T^{2} \)
37 \( 1 + (6 + 6i)T + 37iT^{2} \)
41 \( 1 + 9.48iT - 41T^{2} \)
43 \( 1 + (-3 + 3i)T - 43iT^{2} \)
47 \( 1 + (4.74 - 4.74i)T - 47iT^{2} \)
53 \( 1 + (-1 + i)T - 53iT^{2} \)
59 \( 1 - 9.48T + 59T^{2} \)
61 \( 1 - 6.32iT - 61T^{2} \)
67 \( 1 + (-1 - i)T + 67iT^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 13iT - 79T^{2} \)
83 \( 1 + (3.16 + 3.16i)T + 83iT^{2} \)
89 \( 1 - 6.32T + 89T^{2} \)
97 \( 1 + (-1.58 + 1.58i)T - 97iT^{2} \)
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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

−10.61860759133492127799939088917, −10.00339901221551453740911806678, −9.222517401196385811829025710929, −8.799173275732201099688846565933, −7.34429990007041802233415763209, −6.42672862580717191547418087734, −5.50877598818272985136276309381, −3.94235624711033108667350660203, −3.31094172158265691858067997645, −2.17170321123647421157774710098, 1.11589356614404738373358188247, 2.43321540535345335677756445630, 3.48979161712265632384259784247, 4.97456812635753227234560193581, 6.21015018296273942893591595611, 6.89928301812361060258880972407, 8.020603616713042134092112816919, 8.706420125629455490600368246779, 9.558179780345148727497139457429, 10.23834185381862191790742243387

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