Properties

Label 2-17e2-1.1-c3-0-52
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 $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.15·2-s − 1.99·3-s + 1.94·4-s + 5.00·5-s − 6.27·6-s − 2.78·7-s − 19.1·8-s − 23.0·9-s + 15.7·10-s + 27.2·11-s − 3.86·12-s − 59.7·13-s − 8.78·14-s − 9.97·15-s − 75.7·16-s − 72.6·18-s + 33.2·19-s + 9.71·20-s + 5.54·21-s + 85.9·22-s − 210.·23-s + 38.0·24-s − 99.9·25-s − 188.·26-s + 99.6·27-s − 5.40·28-s + 20.0·29-s + ⋯
L(s)  = 1  + 1.11·2-s − 0.383·3-s + 0.242·4-s + 0.447·5-s − 0.427·6-s − 0.150·7-s − 0.844·8-s − 0.853·9-s + 0.499·10-s + 0.746·11-s − 0.0929·12-s − 1.27·13-s − 0.167·14-s − 0.171·15-s − 1.18·16-s − 0.951·18-s + 0.401·19-s + 0.108·20-s + 0.0576·21-s + 0.832·22-s − 1.91·23-s + 0.323·24-s − 0.799·25-s − 1.42·26-s + 0.710·27-s − 0.0364·28-s + 0.128·29-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: \(1\)
Selberg data: \((2,\ 289,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.15T + 8T^{2} \)
3 \( 1 + 1.99T + 27T^{2} \)
5 \( 1 - 5.00T + 125T^{2} \)
7 \( 1 + 2.78T + 343T^{2} \)
11 \( 1 - 27.2T + 1.33e3T^{2} \)
13 \( 1 + 59.7T + 2.19e3T^{2} \)
19 \( 1 - 33.2T + 6.85e3T^{2} \)
23 \( 1 + 210.T + 1.21e4T^{2} \)
29 \( 1 - 20.0T + 2.43e4T^{2} \)
31 \( 1 + 133.T + 2.97e4T^{2} \)
37 \( 1 - 152.T + 5.06e4T^{2} \)
41 \( 1 - 261.T + 6.89e4T^{2} \)
43 \( 1 + 316.T + 7.95e4T^{2} \)
47 \( 1 - 329.T + 1.03e5T^{2} \)
53 \( 1 + 310.T + 1.48e5T^{2} \)
59 \( 1 + 54.7T + 2.05e5T^{2} \)
61 \( 1 + 818.T + 2.26e5T^{2} \)
67 \( 1 - 731.T + 3.00e5T^{2} \)
71 \( 1 - 629.T + 3.57e5T^{2} \)
73 \( 1 - 496.T + 3.89e5T^{2} \)
79 \( 1 + 90.0T + 4.93e5T^{2} \)
83 \( 1 - 364.T + 5.71e5T^{2} \)
89 \( 1 + 192.T + 7.04e5T^{2} \)
97 \( 1 - 1.35e3T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.30951614177661253552838372375, −9.884516022335621933124737218581, −9.215835871365767951604292013354, −7.86539954466923170500747977677, −6.41503818185662722563619987387, −5.77044312534752046893108224573, −4.81031335352482210923098782440, −3.64662565561276960581137180689, −2.31997875171128384426861282065, 0, 2.31997875171128384426861282065, 3.64662565561276960581137180689, 4.81031335352482210923098782440, 5.77044312534752046893108224573, 6.41503818185662722563619987387, 7.86539954466923170500747977677, 9.215835871365767951604292013354, 9.884516022335621933124737218581, 11.30951614177661253552838372375

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