Properties

Label 2-13e2-1.1-c3-0-31
Degree $2$
Conductor $169$
Sign $-1$
Analytic cond. $9.97132$
Root an. cond. $3.15774$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.72·2-s + 6.89·3-s − 5.01·4-s − 20.8·5-s + 11.9·6-s − 7.56·7-s − 22.4·8-s + 20.5·9-s − 35.9·10-s + 4.40·11-s − 34.5·12-s − 13.0·14-s − 143.·15-s + 1.27·16-s − 73.0·17-s + 35.5·18-s − 55.9·19-s + 104.·20-s − 52.1·21-s + 7.60·22-s − 33.6·23-s − 155.·24-s + 308.·25-s − 44.4·27-s + 37.9·28-s + 121.·29-s − 248.·30-s + ⋯
L(s)  = 1  + 0.610·2-s + 1.32·3-s − 0.626·4-s − 1.86·5-s + 0.810·6-s − 0.408·7-s − 0.993·8-s + 0.761·9-s − 1.13·10-s + 0.120·11-s − 0.831·12-s − 0.249·14-s − 2.47·15-s + 0.0199·16-s − 1.04·17-s + 0.464·18-s − 0.675·19-s + 1.16·20-s − 0.542·21-s + 0.0737·22-s − 0.304·23-s − 1.31·24-s + 2.47·25-s − 0.316·27-s + 0.256·28-s + 0.777·29-s − 1.51·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-1$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ -1)\)

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)$
bad13 \( 1 \)
good2 \( 1 - 1.72T + 8T^{2} \)
3 \( 1 - 6.89T + 27T^{2} \)
5 \( 1 + 20.8T + 125T^{2} \)
7 \( 1 + 7.56T + 343T^{2} \)
11 \( 1 - 4.40T + 1.33e3T^{2} \)
17 \( 1 + 73.0T + 4.91e3T^{2} \)
19 \( 1 + 55.9T + 6.85e3T^{2} \)
23 \( 1 + 33.6T + 1.21e4T^{2} \)
29 \( 1 - 121.T + 2.43e4T^{2} \)
31 \( 1 - 84.1T + 2.97e4T^{2} \)
37 \( 1 + 171.T + 5.06e4T^{2} \)
41 \( 1 + 93.5T + 6.89e4T^{2} \)
43 \( 1 - 441.T + 7.95e4T^{2} \)
47 \( 1 + 272.T + 1.03e5T^{2} \)
53 \( 1 + 480.T + 1.48e5T^{2} \)
59 \( 1 + 350.T + 2.05e5T^{2} \)
61 \( 1 + 484.T + 2.26e5T^{2} \)
67 \( 1 - 967.T + 3.00e5T^{2} \)
71 \( 1 - 402.T + 3.57e5T^{2} \)
73 \( 1 - 351.T + 3.89e5T^{2} \)
79 \( 1 + 820.T + 4.93e5T^{2} \)
83 \( 1 + 192.T + 5.71e5T^{2} \)
89 \( 1 - 813.T + 7.04e5T^{2} \)
97 \( 1 + 788.T + 9.12e5T^{2} \)
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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

−12.16812455910547595100726160925, −11.00781766711374580678395097200, −9.444199515001512156907957427716, −8.533501220661045299739017416930, −7.996697905250206589264176345475, −6.64681847461118279845310507779, −4.60553623703116257751245642097, −3.83990001373111596933014957955, −2.93564508222258030457684441275, 0, 2.93564508222258030457684441275, 3.83990001373111596933014957955, 4.60553623703116257751245642097, 6.64681847461118279845310507779, 7.996697905250206589264176345475, 8.533501220661045299739017416930, 9.444199515001512156907957427716, 11.00781766711374580678395097200, 12.16812455910547595100726160925

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