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Properties

Label 2-2520-7.2-c1-0-10
Degree $2$
Conductor $2520$
Sign $0.749 - 0.661i$
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 + (−2.62 + 0.358i)7-s + (−2.41 − 4.18i)11-s + 2·13-s + (1.82 + 3.16i)17-s + (−2.82 + 4.89i)19-s + (−4.20 + 7.28i)23-s + (−0.499 − 0.866i)25-s + 2.17·29-s + (2.41 + 4.18i)31-s + (−1 + 2.44i)35-s + (2.82 − 4.89i)37-s − 0.171·41-s + 12.8·43-s + (0.171 − 0.297i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.990 + 0.135i)7-s + (−0.727 − 1.26i)11-s + 0.554·13-s + (0.443 + 0.768i)17-s + (−0.648 + 1.12i)19-s + (−0.877 + 1.51i)23-s + (−0.0999 − 0.173i)25-s + 0.403·29-s + (0.433 + 0.751i)31-s + (−0.169 + 0.414i)35-s + (0.464 − 0.805i)37-s − 0.0267·41-s + 1.96·43-s + (0.0250 − 0.0433i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.749 - 0.661i$
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: \((2,\ 2520,\ (\ :1/2),\ 0.749 - 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.327264951\)
\(L(\frac12)\) \(\approx\) \(1.327264951\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 - 0.358i)T \)
good11 \( 1 + (2.41 + 4.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-1.82 - 3.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.82 - 4.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.20 - 7.28i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.17T + 29T^{2} \)
31 \( 1 + (-2.41 - 4.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.82 + 4.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.171T + 41T^{2} \)
43 \( 1 - 12.8T + 43T^{2} \)
47 \( 1 + (-0.171 + 0.297i)T + (-23.5 - 40.7i)T^{2} \)
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 \( 1 + (-2.32 + 4.03i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.44 + 5.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-3.82 - 6.63i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + (8.32 - 14.4i)T + (-44.5 - 77.0i)T^{2} \)
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

−9.011067256360728508789808257896, −8.191341045582786787174626713082, −7.72061773148124464095115178363, −6.38210506206483103841455015678, −5.93555578692099826545490245355, −5.36484403432371547884765950206, −3.91788069900409650982774646687, −3.45000419729918021257313063469, −2.29200634848773662702430083417, −0.970592377212212308577829135201, 0.53482029444019330684399405095, 2.33216782280461266691121172318, 2.80425934696759322116782140099, 4.08928388900415759221756013795, 4.75909000353120519273472846408, 5.88398197181090572024400949392, 6.56345932477489064408644240722, 7.18041928776820985954413129103, 7.984550729997755448293350798376, 8.927860894239900708334092878782

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