Properties

Label 2-2520-7.4-c1-0-11
Degree $2$
Conductor $2520$
Sign $-0.0725 - 0.997i$
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 + (1.62 + 2.09i)7-s + (0.414 − 0.717i)11-s + 2·13-s + (−3.82 + 6.63i)17-s + (2.82 + 4.89i)19-s + (−2.79 − 4.83i)23-s + (−0.499 + 0.866i)25-s + 7.82·29-s + (−0.414 + 0.717i)31-s + (−0.999 + 2.44i)35-s + (−2.82 − 4.89i)37-s − 5.82·41-s − 6.89·43-s + (5.82 + 10.0i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (0.612 + 0.790i)7-s + (0.124 − 0.216i)11-s + 0.554·13-s + (−0.928 + 1.60i)17-s + (0.648 + 1.12i)19-s + (−0.582 − 1.00i)23-s + (−0.0999 + 0.173i)25-s + 1.45·29-s + (−0.0743 + 0.128i)31-s + (−0.169 + 0.414i)35-s + (−0.464 − 0.805i)37-s − 0.910·41-s − 1.05·43-s + (0.850 + 1.47i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0725 - 0.997i$
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: \((2,\ 2520,\ (\ :1/2),\ -0.0725 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.821031740\)
\(L(\frac12)\) \(\approx\) \(1.821031740\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 2.09i)T \)
good11 \( 1 + (-0.414 + 0.717i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (3.82 - 6.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.82 - 4.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.79 + 4.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.82T + 29T^{2} \)
31 \( 1 + (0.414 - 0.717i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.82 + 4.89i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.82T + 41T^{2} \)
43 \( 1 + 6.89T + 43T^{2} \)
47 \( 1 + (-5.82 - 10.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.82 - 4.89i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.32 + 5.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.44 + 11.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (1.82 - 3.16i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.75T + 83T^{2} \)
89 \( 1 + (2.67 + 4.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−8.929173594889441164153987075993, −8.350692596481165551554442001354, −7.83453003234661703796460138042, −6.46481010721885819256226137736, −6.21558378262829470543624204442, −5.27890967864433022038349358428, −4.30739923653229739574235402843, −3.42557380897991546346228384709, −2.30563089604426245328236860286, −1.46180536630652390261111806603, 0.61816942632968231731532398729, 1.71576116553238471868079098067, 2.90282271864640098044493308804, 3.97658339788989692255532162868, 4.88727134231642076785217797898, 5.28296473569994512835047054949, 6.71314728918003610270718952679, 7.00085642845096279888259115386, 8.028094556411129383380650215997, 8.677605853438455799626690998280

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