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Properties

Label 2-2520-7.2-c1-0-23
Degree $2$
Conductor $2520$
Sign $0.701 + 0.712i$
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 + (2 + 3.46i)11-s − 5·13-s + (3 + 5.19i)17-s + (2.5 − 4.33i)19-s + (1 − 1.73i)23-s + (−0.499 − 0.866i)25-s + 2·29-s + (4.5 + 7.79i)31-s + (−2.5 − 0.866i)35-s + (5.5 − 9.52i)37-s + 8·41-s + 43-s + (3 − 5.19i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.188 − 0.981i)7-s + (0.603 + 1.04i)11-s − 1.38·13-s + (0.727 + 1.26i)17-s + (0.573 − 0.993i)19-s + (0.208 − 0.361i)23-s + (−0.0999 − 0.173i)25-s + 0.371·29-s + (0.808 + 1.39i)31-s + (−0.422 − 0.146i)35-s + (0.904 − 1.56i)37-s + 1.24·41-s + 0.152·43-s + (0.437 − 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.701 + 0.712i$
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: \((2,\ 2520,\ (\ :1/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.813637316\)
\(L(\frac12)\) \(\approx\) \(1.813637316\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (-4.5 - 7.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.5 + 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (-9 + 15.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 18T + 97T^{2} \)
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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

−8.945500486712343537465878625794, −7.916401975286515566285235869908, −7.22968603949797022611143004852, −6.72042549731724403430717921221, −5.64795309757959874128509877582, −4.67822718146505756568354479793, −4.22004423089725902282729164698, −3.04635900313064856331066064346, −1.90476273005603457859720657343, −0.74695568084358125539610465876, 1.03107015366364410136875843680, 2.60863354991612477251737618721, 2.95557224206179380213546575652, 4.21192247372831213276148680650, 5.30497190873949320518899625233, 5.84349386158262039328615542371, 6.60829075546506675486776675896, 7.59052378947856956372143734361, 8.130786465887345410829399055650, 9.273162377825351207644446445699

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