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Properties

Label 2-2520-7.4-c1-0-10
Degree $2$
Conductor $2520$
Sign $0.386 - 0.922i$
Analytic cond. $20.1223$
Root an. cond. $4.48578$
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  + (−0.5 − 0.866i)5-s + (0.5 + 2.59i)7-s + (−1 + 1.73i)11-s + 4·13-s + (−3 − 5.19i)19-s + (1.5 + 2.59i)23-s + (−0.499 + 0.866i)25-s + 3·29-s + (2 − 1.73i)35-s + (6 + 10.3i)37-s + 7·41-s − 9·43-s + (−6.5 + 2.59i)49-s + (−3 + 5.19i)53-s + 1.99·55-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (0.188 + 0.981i)7-s + (−0.301 + 0.522i)11-s + 1.10·13-s + (−0.688 − 1.19i)19-s + (0.312 + 0.541i)23-s + (−0.0999 + 0.173i)25-s + 0.557·29-s + (0.338 − 0.292i)35-s + (0.986 + 1.70i)37-s + 1.09·41-s − 1.37·43-s + (−0.928 + 0.371i)49-s + (−0.412 + 0.713i)53-s + 0.269·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(20.1223\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.576231763\)
\(L(\frac12)\) \(\approx\) \(1.576231763\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 2.59i)T \)
good11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6 - 10.3i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7T + 41T^{2} \)
43 \( 1 + 9T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (-4 + 6.92i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3 + 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3T + 83T^{2} \)
89 \( 1 + (-8.5 - 14.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−8.985348379262598672160624337677, −8.366554967923901525035597744474, −7.70115667696930569735549798262, −6.61713571172532146167780417948, −6.01256469646469332411817568920, −5.00262717469344381907404739016, −4.47973933341016504090492763878, −3.26066791083084854413452776095, −2.35904213565899716276703026092, −1.17869253415816511076658667291, 0.58823867328856683973835686736, 1.85179670190566684728763055076, 3.18692111448059849668400841594, 3.87513589782441008584409544482, 4.64419097857425265816712528901, 5.85653184168486094299171562540, 6.40533711741327095924847385374, 7.29234291620048832883936117670, 8.079777311290950633962020751718, 8.522310125162418210689562502990

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