Properties

Label 2-660-1.1-c5-0-15
Degree $2$
Conductor $660$
Sign $1$
Analytic cond. $105.853$
Root an. cond. $10.2885$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s − 25·5-s + 188·7-s + 81·9-s + 121·11-s + 698·13-s − 225·15-s + 1.89e3·17-s − 2.42e3·19-s + 1.69e3·21-s + 2.34e3·23-s + 625·25-s + 729·27-s + 990·29-s + 128·31-s + 1.08e3·33-s − 4.70e3·35-s − 3.20e3·37-s + 6.28e3·39-s + 1.73e4·41-s − 6.65e3·43-s − 2.02e3·45-s − 2.50e4·47-s + 1.85e4·49-s + 1.70e4·51-s − 1.82e4·53-s − 3.02e3·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1.45·7-s + 1/3·9-s + 0.301·11-s + 1.14·13-s − 0.258·15-s + 1.58·17-s − 1.54·19-s + 0.837·21-s + 0.922·23-s + 1/5·25-s + 0.192·27-s + 0.218·29-s + 0.0239·31-s + 0.174·33-s − 0.648·35-s − 0.384·37-s + 0.661·39-s + 1.61·41-s − 0.548·43-s − 0.149·45-s − 1.65·47-s + 1.10·49-s + 0.915·51-s − 0.892·53-s − 0.134·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(660\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 11\)
Sign: $1$
Analytic conductor: \(105.853\)
Root analytic conductor: \(10.2885\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: $\chi_{660} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 660,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.684456021\)
\(L(\frac12)\) \(\approx\) \(3.684456021\)
\(L(\frac{7}{2})\) 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 \( 1 - p^{2} T \)
5 \( 1 + p^{2} T \)
11 \( 1 - p^{2} T \)
good7 \( 1 - 188 T + p^{5} T^{2} \)
13 \( 1 - 698 T + p^{5} T^{2} \)
17 \( 1 - 1890 T + p^{5} T^{2} \)
19 \( 1 + 2428 T + p^{5} T^{2} \)
23 \( 1 - 2340 T + p^{5} T^{2} \)
29 \( 1 - 990 T + p^{5} T^{2} \)
31 \( 1 - 128 T + p^{5} T^{2} \)
37 \( 1 + 3202 T + p^{5} T^{2} \)
41 \( 1 - 17370 T + p^{5} T^{2} \)
43 \( 1 + 6652 T + p^{5} T^{2} \)
47 \( 1 + 25020 T + p^{5} T^{2} \)
53 \( 1 + 18246 T + p^{5} T^{2} \)
59 \( 1 + 8652 T + p^{5} T^{2} \)
61 \( 1 - 37682 T + p^{5} T^{2} \)
67 \( 1 + 18676 T + p^{5} T^{2} \)
71 \( 1 + 2340 T + p^{5} T^{2} \)
73 \( 1 - 43058 T + p^{5} T^{2} \)
79 \( 1 - 65300 T + p^{5} T^{2} \)
83 \( 1 + 55308 T + p^{5} T^{2} \)
89 \( 1 + 64806 T + p^{5} T^{2} \)
97 \( 1 - 38306 T + p^{5} T^{2} \)
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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

−9.689931215296000954282837832008, −8.513981243848699639414765742293, −8.269827549253561423024400083523, −7.35567634279650824260029021365, −6.22541400457682266545682749625, −5.04579607711984970655295089809, −4.15616509142146574176897425504, −3.21913728522344074149345842608, −1.82309285761218339648562654930, −0.968559901238569497163450775928, 0.968559901238569497163450775928, 1.82309285761218339648562654930, 3.21913728522344074149345842608, 4.15616509142146574176897425504, 5.04579607711984970655295089809, 6.22541400457682266545682749625, 7.35567634279650824260029021365, 8.269827549253561423024400083523, 8.513981243848699639414765742293, 9.689931215296000954282837832008

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