Properties

Label 2-90e2-5.4-c1-0-57
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  + 0.732i·7-s + 1.73·11-s + 1.46i·13-s + 1.26i·17-s − 2.46·19-s − 3.46i·23-s − 4.26·29-s − 7.92·31-s + 4.19i·37-s + 0.803·41-s − 6.73i·43-s + 4.73i·47-s + 6.46·49-s − 10.7i·53-s − 4.26·59-s + ⋯
L(s)  = 1  + 0.276i·7-s + 0.522·11-s + 0.406i·13-s + 0.307i·17-s − 0.565·19-s − 0.722i·23-s − 0.792·29-s − 1.42·31-s + 0.689i·37-s + 0.125·41-s − 1.02i·43-s + 0.690i·47-s + 0.923·49-s − 1.47i·53-s − 0.555·59-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

\(L(1)\) \(\approx\) \(0.8043169040\)
\(L(\frac12)\) \(\approx\) \(0.8043169040\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.732iT - 7T^{2} \)
11 \( 1 - 1.73T + 11T^{2} \)
13 \( 1 - 1.46iT - 13T^{2} \)
17 \( 1 - 1.26iT - 17T^{2} \)
19 \( 1 + 2.46T + 19T^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
29 \( 1 + 4.26T + 29T^{2} \)
31 \( 1 + 7.92T + 31T^{2} \)
37 \( 1 - 4.19iT - 37T^{2} \)
41 \( 1 - 0.803T + 41T^{2} \)
43 \( 1 + 6.73iT - 43T^{2} \)
47 \( 1 - 4.73iT - 47T^{2} \)
53 \( 1 + 10.7iT - 53T^{2} \)
59 \( 1 + 4.26T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + 14.3iT - 67T^{2} \)
71 \( 1 - 0.803T + 71T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 + 6.39T + 79T^{2} \)
83 \( 1 - 9.12iT - 83T^{2} \)
89 \( 1 + 5.19T + 89T^{2} \)
97 \( 1 + 2.73iT - 97T^{2} \)
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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

−7.58532059220180528423111794867, −6.85319282589035744324347460079, −6.26707533240463514297015927530, −5.57102585267548162113678575558, −4.76016505369932647820947080055, −4.00897604053955738836722710170, −3.34179194150733243741301246630, −2.25194070438566794164048200024, −1.59060936046664083177389947721, −0.19290917447958255108560666359, 1.08561613217301888949976881122, 2.02333429146801692597178282020, 2.99951132303552983284904893186, 3.83965080749752124721595933566, 4.38056870104588391142390359373, 5.43373247788282536580030267928, 5.84684302046422547312797541915, 6.76843687147166538265192234201, 7.40601667554485144387382239761, 7.86419007211873361848351230756

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