Properties

Label 2-17661-1.1-c1-0-5
Degree $2$
Conductor $17661$
Sign $-1$
Analytic cond. $141.023$
Root an. cond. $11.8753$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 4-s − 3·5-s − 6-s + 7-s + 3·8-s + 9-s + 3·10-s + 6·11-s − 12-s − 6·13-s − 14-s − 3·15-s − 16-s − 17-s − 18-s + 6·19-s + 3·20-s + 21-s − 6·22-s + 6·23-s + 3·24-s + 4·25-s + 6·26-s + 27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.948·10-s + 1.80·11-s − 0.288·12-s − 1.66·13-s − 0.267·14-s − 0.774·15-s − 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.37·19-s + 0.670·20-s + 0.218·21-s − 1.27·22-s + 1.25·23-s + 0.612·24-s + 4/5·25-s + 1.17·26-s + 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17661 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17661 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17661\)    =    \(3 \cdot 7 \cdot 29^{2}\)
Sign: $-1$
Analytic conductor: \(141.023\)
Root analytic conductor: \(11.8753\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 17661,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 - T \)
29 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−16.38808451100912, −15.33612446962717, −14.94408679964406, −14.70380073178517, −13.84826962625982, −13.62378814994830, −12.64592287768261, −12.12618412733689, −11.52759362501634, −11.36128477132887, −10.23096471564520, −9.746544633216211, −9.354067503967411, −8.580406803591663, −8.349017125213281, −7.589175040037986, −7.081207860207030, −6.792223529995703, −5.352220214682454, −4.682026531343758, −4.344681123606651, −3.518671407075754, −2.983820556084452, −1.694344503381537, −1.035581937693167, 0, 1.035581937693167, 1.694344503381537, 2.983820556084452, 3.518671407075754, 4.344681123606651, 4.682026531343758, 5.352220214682454, 6.792223529995703, 7.081207860207030, 7.589175040037986, 8.349017125213281, 8.580406803591663, 9.354067503967411, 9.746544633216211, 10.23096471564520, 11.36128477132887, 11.52759362501634, 12.12618412733689, 12.64592287768261, 13.62378814994830, 13.84826962625982, 14.70380073178517, 14.94408679964406, 15.33612446962717, 16.38808451100912

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