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Properties

Label 2-2520-7.2-c1-0-1
Degree $2$
Conductor $2520$
Sign $-0.642 - 0.766i$
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 + (−0.292 − 2.62i)7-s + (0.839 + 1.45i)11-s − 4.84·13-s + (1 + 1.73i)17-s + (−3.42 + 5.92i)19-s + (−1.13 + 1.95i)23-s + (−0.499 − 0.866i)25-s − 3.32·29-s + (−4.58 − 7.94i)31-s + (−2.42 − 1.06i)35-s + (1.42 − 2.46i)37-s − 9.52·41-s + 6.58·43-s + (−6.10 + 10.5i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.110 − 0.993i)7-s + (0.253 + 0.438i)11-s − 1.34·13-s + (0.242 + 0.420i)17-s + (−0.785 + 1.36i)19-s + (−0.235 + 0.408i)23-s + (−0.0999 − 0.173i)25-s − 0.616·29-s + (−0.823 − 1.42i)31-s + (−0.409 − 0.179i)35-s + (0.233 − 0.405i)37-s − 1.48·41-s + 1.00·43-s + (−0.890 + 1.54i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(20.1223\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :1/2),\ -0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3929544129\)
\(L(\frac12)\) \(\approx\) \(0.3929544129\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.292 + 2.62i)T \)
good11 \( 1 + (-0.839 - 1.45i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.84T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.42 - 5.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.13 - 1.95i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.32T + 29T^{2} \)
31 \( 1 + (4.58 + 7.94i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.42 + 2.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.52T + 41T^{2} \)
43 \( 1 - 6.58T + 43T^{2} \)
47 \( 1 + (6.10 - 10.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.74 - 6.48i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.24 + 5.62i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.87 - 4.98i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-5.84 - 10.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.84 - 4.93i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 + (2.92 - 5.06i)T + (-44.5 - 77.0i)T^{2} \)
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

−9.495507804450190335390939940344, −8.255199143380914169656143979615, −7.66476746719832208536965904499, −7.01344091563381378776527095074, −6.06173221284233243280282578066, −5.29641825386446885770378738738, −4.24943467309685448463013669456, −3.80419692757467338553021648855, −2.37130373817640571469190013112, −1.41082642898021284074276251988, 0.12270912413678648168391418679, 1.99252110194495123582424826208, 2.69670938809675836177630968169, 3.61265636295704168427743753948, 4.99382670101138092638095563629, 5.29307505090403927470189853537, 6.53985584633104026765770581858, 6.87026004933014604854147792478, 7.918295456186772326615584974110, 8.791165728989770086550239206906

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