Properties

Label 2-663-1.1-c3-0-93
Degree $2$
Conductor $663$
Sign $-1$
Analytic cond. $39.1182$
Root an. cond. $6.25445$
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  + 4·2-s + 3·3-s + 8·4-s − 10·5-s + 12·6-s − 10·7-s + 9·9-s − 40·10-s + 18·11-s + 24·12-s − 13·13-s − 40·14-s − 30·15-s − 64·16-s − 17·17-s + 36·18-s − 74·19-s − 80·20-s − 30·21-s + 72·22-s − 132·23-s − 25·25-s − 52·26-s + 27·27-s − 80·28-s + 210·29-s − 120·30-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s − 0.894·5-s + 0.816·6-s − 0.539·7-s + 1/3·9-s − 1.26·10-s + 0.493·11-s + 0.577·12-s − 0.277·13-s − 0.763·14-s − 0.516·15-s − 16-s − 0.242·17-s + 0.471·18-s − 0.893·19-s − 0.894·20-s − 0.311·21-s + 0.697·22-s − 1.19·23-s − 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.539·28-s + 1.34·29-s − 0.730·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(663\)    =    \(3 \cdot 13 \cdot 17\)
Sign: $-1$
Analytic conductor: \(39.1182\)
Root analytic conductor: \(6.25445\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{663} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 663,\ (\ :3/2),\ -1)\)

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)$
bad3 \( 1 - p T \)
13 \( 1 + p T \)
17 \( 1 + p T \)
good2 \( 1 - p^{2} T + p^{3} T^{2} \)
5 \( 1 + 2 p T + p^{3} T^{2} \)
7 \( 1 + 10 T + p^{3} T^{2} \)
11 \( 1 - 18 T + p^{3} T^{2} \)
19 \( 1 + 74 T + p^{3} T^{2} \)
23 \( 1 + 132 T + p^{3} T^{2} \)
29 \( 1 - 210 T + p^{3} T^{2} \)
31 \( 1 + 230 T + p^{3} T^{2} \)
37 \( 1 + 46 T + p^{3} T^{2} \)
41 \( 1 + 114 T + p^{3} T^{2} \)
43 \( 1 - 36 T + p^{3} T^{2} \)
47 \( 1 - 446 T + p^{3} T^{2} \)
53 \( 1 + 754 T + p^{3} T^{2} \)
59 \( 1 + 50 T + p^{3} T^{2} \)
61 \( 1 + 226 T + p^{3} T^{2} \)
67 \( 1 - 582 T + p^{3} T^{2} \)
71 \( 1 + 370 T + p^{3} T^{2} \)
73 \( 1 - 826 T + p^{3} T^{2} \)
79 \( 1 - 272 T + p^{3} T^{2} \)
83 \( 1 - 162 T + p^{3} T^{2} \)
89 \( 1 + 186 T + p^{3} T^{2} \)
97 \( 1 + 790 T + p^{3} T^{2} \)
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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

−9.659843976790960064473631130333, −8.754217380540816521227387513871, −7.82104790607084610911641564450, −6.78579658254480417848825680762, −6.05890952920649466166152247626, −4.75069883847733021473568200974, −3.98048430488054591144503268863, −3.33533195776516530491471005583, −2.15428215019628625942117917398, 0, 2.15428215019628625942117917398, 3.33533195776516530491471005583, 3.98048430488054591144503268863, 4.75069883847733021473568200974, 6.05890952920649466166152247626, 6.78579658254480417848825680762, 7.82104790607084610911641564450, 8.754217380540816521227387513871, 9.659843976790960064473631130333

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