Properties

Label 2-13e2-1.1-c3-0-16
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.56·2-s + 8.68·3-s − 5.56·4-s + 3.56·5-s + 13.5·6-s + 27.1·7-s − 21.1·8-s + 48.4·9-s + 5.56·10-s − 15.2·11-s − 48.3·12-s + 42.4·14-s + 30.9·15-s + 11.4·16-s + 44.5·17-s + 75.6·18-s − 23.9·19-s − 19.8·20-s + 236.·21-s − 23.8·22-s + 122.·23-s − 183.·24-s − 112.·25-s + 186.·27-s − 151.·28-s − 219.·29-s + 48.3·30-s + ⋯
L(s)  = 1  + 0.552·2-s + 1.67·3-s − 0.695·4-s + 0.318·5-s + 0.922·6-s + 1.46·7-s − 0.935·8-s + 1.79·9-s + 0.175·10-s − 0.418·11-s − 1.16·12-s + 0.810·14-s + 0.532·15-s + 0.178·16-s + 0.635·17-s + 0.990·18-s − 0.289·19-s − 0.221·20-s + 2.45·21-s − 0.230·22-s + 1.11·23-s − 1.56·24-s − 0.898·25-s + 1.32·27-s − 1.02·28-s − 1.40·29-s + 0.293·30-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: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.647600146\)
\(L(\frac12)\) \(\approx\) \(3.647600146\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.56T + 8T^{2} \)
3 \( 1 - 8.68T + 27T^{2} \)
5 \( 1 - 3.56T + 125T^{2} \)
7 \( 1 - 27.1T + 343T^{2} \)
11 \( 1 + 15.2T + 1.33e3T^{2} \)
17 \( 1 - 44.5T + 4.91e3T^{2} \)
19 \( 1 + 23.9T + 6.85e3T^{2} \)
23 \( 1 - 122.T + 1.21e4T^{2} \)
29 \( 1 + 219.T + 2.43e4T^{2} \)
31 \( 1 + 27.0T + 2.97e4T^{2} \)
37 \( 1 + 94.1T + 5.06e4T^{2} \)
41 \( 1 - 160.T + 6.89e4T^{2} \)
43 \( 1 + 151.T + 7.95e4T^{2} \)
47 \( 1 + 466.T + 1.03e5T^{2} \)
53 \( 1 + 120.T + 1.48e5T^{2} \)
59 \( 1 - 439.T + 2.05e5T^{2} \)
61 \( 1 + 137.T + 2.26e5T^{2} \)
67 \( 1 + 512.T + 3.00e5T^{2} \)
71 \( 1 + 410.T + 3.57e5T^{2} \)
73 \( 1 - 308.T + 3.89e5T^{2} \)
79 \( 1 + 586.T + 4.93e5T^{2} \)
83 \( 1 + 1.35e3T + 5.71e5T^{2} \)
89 \( 1 + 439.T + 7.04e5T^{2} \)
97 \( 1 - 1.51e3T + 9.12e5T^{2} \)
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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

−12.85327148461719649280755181209, −11.44032636412743274945387930294, −10.02033775163800621410418605343, −9.064744432115461292949648961231, −8.296230184723724804150694207678, −7.50191130115328308288265967836, −5.47336843464949542583186927761, −4.40149866017920026425507641315, −3.21871588760171694290290033204, −1.78182775320839772477496692489, 1.78182775320839772477496692489, 3.21871588760171694290290033204, 4.40149866017920026425507641315, 5.47336843464949542583186927761, 7.50191130115328308288265967836, 8.296230184723724804150694207678, 9.064744432115461292949648961231, 10.02033775163800621410418605343, 11.44032636412743274945387930294, 12.85327148461719649280755181209

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