Properties

Label 2-660-1.1-c3-0-14
Degree $2$
Conductor $660$
Sign $-1$
Analytic cond. $38.9412$
Root an. cond. $6.24029$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 5·5-s + 9·9-s + 11·11-s − 42·13-s − 15·15-s − 14·17-s − 52·19-s + 96·23-s + 25·25-s − 27·27-s − 26·29-s − 144·31-s − 33·33-s + 126·37-s + 126·39-s + 58·41-s + 364·43-s + 45·45-s − 328·47-s − 343·49-s + 42·51-s − 50·53-s + 55·55-s + 156·57-s − 284·59-s − 794·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s − 0.896·13-s − 0.258·15-s − 0.199·17-s − 0.627·19-s + 0.870·23-s + 1/5·25-s − 0.192·27-s − 0.166·29-s − 0.834·31-s − 0.174·33-s + 0.559·37-s + 0.517·39-s + 0.220·41-s + 1.29·43-s + 0.149·45-s − 1.01·47-s − 49-s + 0.115·51-s − 0.129·53-s + 0.134·55-s + 0.362·57-s − 0.626·59-s − 1.66·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(660\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 11\)
Sign: $-1$
Analytic conductor: \(38.9412\)
Root analytic conductor: \(6.24029\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{660} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 660,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
5 \( 1 - p T \)
11 \( 1 - p T \)
good7 \( 1 + p^{3} T^{2} \)
13 \( 1 + 42 T + p^{3} T^{2} \)
17 \( 1 + 14 T + p^{3} T^{2} \)
19 \( 1 + 52 T + p^{3} T^{2} \)
23 \( 1 - 96 T + p^{3} T^{2} \)
29 \( 1 + 26 T + p^{3} T^{2} \)
31 \( 1 + 144 T + p^{3} T^{2} \)
37 \( 1 - 126 T + p^{3} T^{2} \)
41 \( 1 - 58 T + p^{3} T^{2} \)
43 \( 1 - 364 T + p^{3} T^{2} \)
47 \( 1 + 328 T + p^{3} T^{2} \)
53 \( 1 + 50 T + p^{3} T^{2} \)
59 \( 1 + 284 T + p^{3} T^{2} \)
61 \( 1 + 794 T + p^{3} T^{2} \)
67 \( 1 + 316 T + p^{3} T^{2} \)
71 \( 1 + 280 T + p^{3} T^{2} \)
73 \( 1 + 358 T + p^{3} T^{2} \)
79 \( 1 - 784 T + p^{3} T^{2} \)
83 \( 1 - 324 T + p^{3} T^{2} \)
89 \( 1 + 1398 T + p^{3} T^{2} \)
97 \( 1 + 894 T + p^{3} T^{2} \)
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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

−9.656905577971047262151725360981, −9.068774202851517893219769020188, −7.82584260910518473904086008208, −6.92066874403063263069664478320, −6.09708839549662697975574164854, −5.13833631927231244764585446237, −4.25653746599888616178700595871, −2.80019645083200556978194164029, −1.52028883848851401307599578277, 0, 1.52028883848851401307599578277, 2.80019645083200556978194164029, 4.25653746599888616178700595871, 5.13833631927231244764585446237, 6.09708839549662697975574164854, 6.92066874403063263069664478320, 7.82584260910518473904086008208, 9.068774202851517893219769020188, 9.656905577971047262151725360981

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