Properties

Label 2-17e2-1.1-c3-0-12
Degree $2$
Conductor $289$
Sign $1$
Analytic cond. $17.0515$
Root an. cond. $4.12935$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.36·2-s − 3.15·3-s − 6.14·4-s − 3.03·5-s − 4.29·6-s + 7.94·7-s − 19.2·8-s − 17.0·9-s − 4.12·10-s − 27.6·11-s + 19.3·12-s + 58.1·13-s + 10.8·14-s + 9.56·15-s + 22.9·16-s − 23.2·18-s + 89.1·19-s + 18.6·20-s − 25.0·21-s − 37.5·22-s + 115.·23-s + 60.7·24-s − 115.·25-s + 79.1·26-s + 138.·27-s − 48.8·28-s + 128.·29-s + ⋯
L(s)  = 1  + 0.481·2-s − 0.607·3-s − 0.768·4-s − 0.271·5-s − 0.292·6-s + 0.428·7-s − 0.851·8-s − 0.631·9-s − 0.130·10-s − 0.756·11-s + 0.466·12-s + 1.23·13-s + 0.206·14-s + 0.164·15-s + 0.358·16-s − 0.303·18-s + 1.07·19-s + 0.208·20-s − 0.260·21-s − 0.364·22-s + 1.04·23-s + 0.516·24-s − 0.926·25-s + 0.596·26-s + 0.990·27-s − 0.329·28-s + 0.823·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $1$
Analytic conductor: \(17.0515\)
Root analytic conductor: \(4.12935\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.254966249\)
\(L(\frac12)\) \(\approx\) \(1.254966249\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 - 1.36T + 8T^{2} \)
3 \( 1 + 3.15T + 27T^{2} \)
5 \( 1 + 3.03T + 125T^{2} \)
7 \( 1 - 7.94T + 343T^{2} \)
11 \( 1 + 27.6T + 1.33e3T^{2} \)
13 \( 1 - 58.1T + 2.19e3T^{2} \)
19 \( 1 - 89.1T + 6.85e3T^{2} \)
23 \( 1 - 115.T + 1.21e4T^{2} \)
29 \( 1 - 128.T + 2.43e4T^{2} \)
31 \( 1 + 273.T + 2.97e4T^{2} \)
37 \( 1 - 132.T + 5.06e4T^{2} \)
41 \( 1 - 470.T + 6.89e4T^{2} \)
43 \( 1 - 352.T + 7.95e4T^{2} \)
47 \( 1 - 152.T + 1.03e5T^{2} \)
53 \( 1 - 527.T + 1.48e5T^{2} \)
59 \( 1 + 292.T + 2.05e5T^{2} \)
61 \( 1 - 53.8T + 2.26e5T^{2} \)
67 \( 1 - 52.9T + 3.00e5T^{2} \)
71 \( 1 + 788.T + 3.57e5T^{2} \)
73 \( 1 + 295.T + 3.89e5T^{2} \)
79 \( 1 - 720.T + 4.93e5T^{2} \)
83 \( 1 + 116.T + 5.71e5T^{2} \)
89 \( 1 + 813.T + 7.04e5T^{2} \)
97 \( 1 + 794.T + 9.12e5T^{2} \)
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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

−11.36151547593780725985562448981, −10.75809049717223082578504584930, −9.376462070625949273247621168080, −8.531408892990994087616245405627, −7.54007051531059185708868274122, −5.92891169100891324104834455317, −5.39385682582968923204369289712, −4.25134514507820346701216094376, −3.03319153829326505427247663103, −0.76398739206330964468500475345, 0.76398739206330964468500475345, 3.03319153829326505427247663103, 4.25134514507820346701216094376, 5.39385682582968923204369289712, 5.92891169100891324104834455317, 7.54007051531059185708868274122, 8.531408892990994087616245405627, 9.376462070625949273247621168080, 10.75809049717223082578504584930, 11.36151547593780725985562448981

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