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Properties

Label 2-90e2-5.4-c1-0-55
Degree $2$
Conductor $8100$
Sign $0.447 + 0.894i$
Analytic cond. $64.6788$
Root an. cond. $8.04231$
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  + 4i·7-s + 3·11-s − 4i·13-s − 5·19-s − 6i·23-s + 9·29-s + 5·31-s − 2i·37-s − 9·41-s − 10i·43-s + 6i·47-s − 9·49-s − 12i·53-s − 9·59-s − 10·61-s + ⋯
L(s)  = 1  + 1.51i·7-s + 0.904·11-s − 1.10i·13-s − 1.14·19-s − 1.25i·23-s + 1.67·29-s + 0.898·31-s − 0.328i·37-s − 1.40·41-s − 1.52i·43-s + 0.875i·47-s − 1.28·49-s − 1.64i·53-s − 1.17·59-s − 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(64.6788\)
Root analytic conductor: \(8.04231\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8100} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8100,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.580075607\)
\(L(\frac12)\) \(\approx\) \(1.580075607\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + 9T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−8.014314369223243594461048982039, −6.60597526108958956137347048781, −6.51472452494962458507393716037, −5.64383531484979410889158488991, −4.95059647334067404023635441222, −4.24118536048522755757380064591, −3.16774684067533091896441644117, −2.57730295356432369557971685450, −1.71680167446038212206007107819, −0.39933647876750581816733663898, 1.05487103233014165842616105893, 1.67560496926417688775667522351, 2.94407829053737253396460231590, 3.81109860777580603617551319637, 4.38374202551815685015196066995, 4.85091576748896371807942654703, 6.23906877904912093020757912163, 6.53239127785860156602342407887, 7.16367256370461133833970837946, 7.896991764601046083267678194026

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