Properties

Label 2-17e2-1.1-c9-0-93
Degree $2$
Conductor $289$
Sign $1$
Analytic cond. $148.845$
Root an. cond. $12.2002$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 38.8·2-s − 39.5·3-s + 1.00e3·4-s − 1.32e3·5-s − 1.53e3·6-s + 2.03e3·7-s + 1.89e4·8-s − 1.81e4·9-s − 5.15e4·10-s + 7.97e4·11-s − 3.95e4·12-s + 1.98e4·13-s + 7.89e4·14-s + 5.25e4·15-s + 2.25e5·16-s − 7.04e5·18-s − 4.30e5·19-s − 1.32e6·20-s − 8.04e4·21-s + 3.10e6·22-s + 1.47e6·23-s − 7.51e5·24-s − 1.93e5·25-s + 7.72e5·26-s + 1.49e6·27-s + 2.03e6·28-s − 5.01e6·29-s + ⋯
L(s)  = 1  + 1.71·2-s − 0.282·3-s + 1.95·4-s − 0.949·5-s − 0.485·6-s + 0.319·7-s + 1.63·8-s − 0.920·9-s − 1.63·10-s + 1.64·11-s − 0.551·12-s + 0.192·13-s + 0.549·14-s + 0.267·15-s + 0.861·16-s − 1.58·18-s − 0.757·19-s − 1.85·20-s − 0.0902·21-s + 2.82·22-s + 1.10·23-s − 0.462·24-s − 0.0989·25-s + 0.331·26-s + 0.542·27-s + 0.624·28-s − 1.31·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $1$
Analytic conductor: \(148.845\)
Root analytic conductor: \(12.2002\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(5.425897408\)
\(L(\frac12)\) \(\approx\) \(5.425897408\)
\(L(\frac{11}{2})\) 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 \( 1 - 38.8T + 512T^{2} \)
3 \( 1 + 39.5T + 1.96e4T^{2} \)
5 \( 1 + 1.32e3T + 1.95e6T^{2} \)
7 \( 1 - 2.03e3T + 4.03e7T^{2} \)
11 \( 1 - 7.97e4T + 2.35e9T^{2} \)
13 \( 1 - 1.98e4T + 1.06e10T^{2} \)
19 \( 1 + 4.30e5T + 3.22e11T^{2} \)
23 \( 1 - 1.47e6T + 1.80e12T^{2} \)
29 \( 1 + 5.01e6T + 1.45e13T^{2} \)
31 \( 1 - 7.99e6T + 2.64e13T^{2} \)
37 \( 1 - 6.88e5T + 1.29e14T^{2} \)
41 \( 1 - 8.23e6T + 3.27e14T^{2} \)
43 \( 1 - 4.37e7T + 5.02e14T^{2} \)
47 \( 1 + 3.64e7T + 1.11e15T^{2} \)
53 \( 1 - 5.12e7T + 3.29e15T^{2} \)
59 \( 1 - 1.34e8T + 8.66e15T^{2} \)
61 \( 1 - 9.39e7T + 1.16e16T^{2} \)
67 \( 1 - 1.29e8T + 2.72e16T^{2} \)
71 \( 1 - 2.39e7T + 4.58e16T^{2} \)
73 \( 1 - 4.14e8T + 5.88e16T^{2} \)
79 \( 1 - 3.87e8T + 1.19e17T^{2} \)
83 \( 1 - 2.80e8T + 1.86e17T^{2} \)
89 \( 1 - 9.07e8T + 3.50e17T^{2} \)
97 \( 1 + 1.62e9T + 7.60e17T^{2} \)
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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

−11.00399255672783189483836666022, −9.194478333010326550562918683046, −8.103542639188138845655945739310, −6.86056474848279616443530014741, −6.17115356936305949498846626170, −5.11749535057259034690094269569, −4.12741156964399267306896990704, −3.54224730084742931530645944661, −2.30668070656791089907476007824, −0.811195267376774017039010584451, 0.811195267376774017039010584451, 2.30668070656791089907476007824, 3.54224730084742931530645944661, 4.12741156964399267306896990704, 5.11749535057259034690094269569, 6.17115356936305949498846626170, 6.86056474848279616443530014741, 8.103542639188138845655945739310, 9.194478333010326550562918683046, 11.00399255672783189483836666022

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