Properties

Label 2-1617-1.1-c1-0-18
Degree $2$
Conductor $1617$
Sign $1$
Analytic cond. $12.9118$
Root an. cond. $3.59330$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.614·2-s + 3-s − 1.62·4-s − 1.49·5-s + 0.614·6-s − 2.22·8-s + 9-s − 0.919·10-s + 11-s − 1.62·12-s + 1.27·13-s − 1.49·15-s + 1.87·16-s − 0.614·17-s + 0.614·18-s + 2.66·19-s + 2.42·20-s + 0.614·22-s + 1.87·23-s − 2.22·24-s − 2.76·25-s + 0.784·26-s + 27-s + 8.44·29-s − 0.919·30-s − 4.62·31-s + 5.60·32-s + ⋯
L(s)  = 1  + 0.434·2-s + 0.577·3-s − 0.810·4-s − 0.669·5-s + 0.251·6-s − 0.787·8-s + 0.333·9-s − 0.290·10-s + 0.301·11-s − 0.468·12-s + 0.353·13-s − 0.386·15-s + 0.468·16-s − 0.149·17-s + 0.144·18-s + 0.610·19-s + 0.542·20-s + 0.131·22-s + 0.390·23-s − 0.454·24-s − 0.552·25-s + 0.153·26-s + 0.192·27-s + 1.56·29-s − 0.167·30-s − 0.830·31-s + 0.991·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(12.9118\)
Root analytic conductor: \(3.59330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1617,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.854422790\)
\(L(\frac12)\) \(\approx\) \(1.854422790\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 - T \)
good2 \( 1 - 0.614T + 2T^{2} \)
5 \( 1 + 1.49T + 5T^{2} \)
13 \( 1 - 1.27T + 13T^{2} \)
17 \( 1 + 0.614T + 17T^{2} \)
19 \( 1 - 2.66T + 19T^{2} \)
23 \( 1 - 1.87T + 23T^{2} \)
29 \( 1 - 8.44T + 29T^{2} \)
31 \( 1 + 4.62T + 31T^{2} \)
37 \( 1 - 0.274T + 37T^{2} \)
41 \( 1 - 4.91T + 41T^{2} \)
43 \( 1 - 9.42T + 43T^{2} \)
47 \( 1 - 3.16T + 47T^{2} \)
53 \( 1 - 6.18T + 53T^{2} \)
59 \( 1 + 1.87T + 59T^{2} \)
61 \( 1 - 8.52T + 61T^{2} \)
67 \( 1 - 5.06T + 67T^{2} \)
71 \( 1 - 16.0T + 71T^{2} \)
73 \( 1 + 8.75T + 73T^{2} \)
79 \( 1 - 9.44T + 79T^{2} \)
83 \( 1 + 9.00T + 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 - 17.6T + 97T^{2} \)
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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

−9.257626056631877390748307134901, −8.627091531296322762873638249057, −7.932629976243291116854589732158, −7.10477496257531470771323476981, −6.04306533835239577131346784769, −5.11290741856533401372807772863, −4.16500731010616285089397301329, −3.65640025009427068687233325851, −2.61580076977099488260277314299, −0.897447477259559899628080971199, 0.897447477259559899628080971199, 2.61580076977099488260277314299, 3.65640025009427068687233325851, 4.16500731010616285089397301329, 5.11290741856533401372807772863, 6.04306533835239577131346784769, 7.10477496257531470771323476981, 7.932629976243291116854589732158, 8.627091531296322762873638249057, 9.257626056631877390748307134901

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