Properties

Label 2-6171-1.1-c1-0-122
Degree $2$
Conductor $6171$
Sign $-1$
Analytic cond. $49.2756$
Root an. cond. $7.01966$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.347·2-s − 3-s − 1.87·4-s − 1.45·5-s − 0.347·6-s − 2.71·7-s − 1.34·8-s + 9-s − 0.506·10-s + 1.87·12-s + 1.83·13-s − 0.944·14-s + 1.45·15-s + 3.28·16-s − 17-s + 0.347·18-s − 5.64·19-s + 2.74·20-s + 2.71·21-s + 2.86·23-s + 1.34·24-s − 2.87·25-s + 0.636·26-s − 27-s + 5.10·28-s − 0.687·29-s + 0.506·30-s + ⋯
L(s)  = 1  + 0.245·2-s − 0.577·3-s − 0.939·4-s − 0.652·5-s − 0.141·6-s − 1.02·7-s − 0.476·8-s + 0.333·9-s − 0.160·10-s + 0.542·12-s + 0.508·13-s − 0.252·14-s + 0.376·15-s + 0.822·16-s − 0.242·17-s + 0.0819·18-s − 1.29·19-s + 0.612·20-s + 0.592·21-s + 0.597·23-s + 0.275·24-s − 0.574·25-s + 0.124·26-s − 0.192·27-s + 0.964·28-s − 0.127·29-s + 0.0925·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6171\)    =    \(3 \cdot 11^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(49.2756\)
Root analytic conductor: \(7.01966\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6171,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
11 \( 1 \)
17 \( 1 + T \)
good2 \( 1 - 0.347T + 2T^{2} \)
5 \( 1 + 1.45T + 5T^{2} \)
7 \( 1 + 2.71T + 7T^{2} \)
13 \( 1 - 1.83T + 13T^{2} \)
19 \( 1 + 5.64T + 19T^{2} \)
23 \( 1 - 2.86T + 23T^{2} \)
29 \( 1 + 0.687T + 29T^{2} \)
31 \( 1 - 4.22T + 31T^{2} \)
37 \( 1 - 9.38T + 37T^{2} \)
41 \( 1 - 0.342T + 41T^{2} \)
43 \( 1 - 0.261T + 43T^{2} \)
47 \( 1 - 3.53T + 47T^{2} \)
53 \( 1 + 7.22T + 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 - 10.7T + 61T^{2} \)
67 \( 1 - 9.18T + 67T^{2} \)
71 \( 1 + 0.296T + 71T^{2} \)
73 \( 1 - 0.0393T + 73T^{2} \)
79 \( 1 + 9.30T + 79T^{2} \)
83 \( 1 - 7.54T + 83T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 - 0.645T + 97T^{2} \)
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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

−7.77616849354285094926980798730, −6.77981438827713594164314512486, −6.25556479446354699925334274771, −5.59684865424052539175401408280, −4.67885773871818093021079361673, −4.06931666519335506289434616596, −3.52797086098515660450031741680, −2.48657754053066598105568525801, −0.901388798758029617951927282359, 0, 0.901388798758029617951927282359, 2.48657754053066598105568525801, 3.52797086098515660450031741680, 4.06931666519335506289434616596, 4.67885773871818093021079361673, 5.59684865424052539175401408280, 6.25556479446354699925334274771, 6.77981438827713594164314512486, 7.77616849354285094926980798730

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