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Properties

Label 2-90-5.4-c15-0-25
Degree $2$
Conductor $90$
Sign $-0.115 - 0.993i$
Analytic cond. $128.424$
Root an. cond. $11.3324$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 128i·2-s − 1.63e4·4-s + (1.73e5 − 2.01e4i)5-s + 2.89e6i·7-s − 2.09e6i·8-s + (2.57e6 + 2.22e7i)10-s + 6.57e7·11-s − 1.16e8i·13-s − 3.70e8·14-s + 2.68e8·16-s + 1.76e9i·17-s + 7.05e9·19-s + (−2.84e9 + 3.29e8i)20-s + 8.41e9i·22-s + 3.96e9i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.993 − 0.115i)5-s + 1.32i·7-s − 0.353i·8-s + (0.0814 + 0.702i)10-s + 1.01·11-s − 0.515i·13-s − 0.938·14-s + 0.250·16-s + 1.04i·17-s + 1.80·19-s + (−0.496 + 0.0576i)20-s + 0.719i·22-s + 0.242i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.115 - 0.993i$
Analytic conductor: \(128.424\)
Root analytic conductor: \(11.3324\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :15/2),\ -0.115 - 0.993i)\)

Particular Values

\(L(8)\) \(\approx\) \(3.160515370\)
\(L(\frac12)\) \(\approx\) \(3.160515370\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 128iT \)
3 \( 1 \)
5 \( 1 + (-1.73e5 + 2.01e4i)T \)
good7 \( 1 - 2.89e6iT - 4.74e12T^{2} \)
11 \( 1 - 6.57e7T + 4.17e15T^{2} \)
13 \( 1 + 1.16e8iT - 5.11e16T^{2} \)
17 \( 1 - 1.76e9iT - 2.86e18T^{2} \)
19 \( 1 - 7.05e9T + 1.51e19T^{2} \)
23 \( 1 - 3.96e9iT - 2.66e20T^{2} \)
29 \( 1 + 6.66e10T + 8.62e21T^{2} \)
31 \( 1 - 5.24e10T + 2.34e22T^{2} \)
37 \( 1 + 5.76e9iT - 3.33e23T^{2} \)
41 \( 1 - 5.49e11T + 1.55e24T^{2} \)
43 \( 1 + 1.67e12iT - 3.17e24T^{2} \)
47 \( 1 + 6.04e12iT - 1.20e25T^{2} \)
53 \( 1 + 3.47e12iT - 7.31e25T^{2} \)
59 \( 1 - 1.41e13T + 3.65e26T^{2} \)
61 \( 1 - 2.55e13T + 6.02e26T^{2} \)
67 \( 1 - 8.85e13iT - 2.46e27T^{2} \)
71 \( 1 - 1.15e14T + 5.87e27T^{2} \)
73 \( 1 + 9.32e13iT - 8.90e27T^{2} \)
79 \( 1 + 1.53e13T + 2.91e28T^{2} \)
83 \( 1 - 2.32e14iT - 6.11e28T^{2} \)
89 \( 1 + 1.36e14T + 1.74e29T^{2} \)
97 \( 1 + 5.63e14iT - 6.33e29T^{2} \)
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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

−11.63802197042360124520431088988, −10.01736685241841317762344559480, −9.204216484375508335559785879206, −8.380711644610606746478302487710, −6.90898423055937483168151305246, −5.77639624230313434726442572634, −5.31860340346644129811360221070, −3.59091812422460538908366142860, −2.18379746886971080593541207685, −1.02592703702595798586058423010, 0.792889020617633769537666828706, 1.43083624370447835001587351815, 2.82753320669268846832314235320, 3.96855784690133492101064722121, 5.10526059867351298492427145230, 6.52741127958041555356660918983, 7.54523808909382858068276011118, 9.319137705329298183500885888262, 9.717618463476404975390563793740, 10.91748730520810411410156657374

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