Properties

Label 2-13e2-1.1-c3-0-13
Degree $2$
Conductor $169$
Sign $1$
Analytic cond. $9.97132$
Root an. cond. $3.15774$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·2-s − 7·3-s + 17·4-s + 7·5-s − 35·6-s + 13·7-s + 45·8-s + 22·9-s + 35·10-s + 26·11-s − 119·12-s + 65·14-s − 49·15-s + 89·16-s + 77·17-s + 110·18-s + 126·19-s + 119·20-s − 91·21-s + 130·22-s − 96·23-s − 315·24-s − 76·25-s + 35·27-s + 221·28-s − 82·29-s − 245·30-s + ⋯
L(s)  = 1  + 1.76·2-s − 1.34·3-s + 17/8·4-s + 0.626·5-s − 2.38·6-s + 0.701·7-s + 1.98·8-s + 0.814·9-s + 1.10·10-s + 0.712·11-s − 2.86·12-s + 1.24·14-s − 0.843·15-s + 1.39·16-s + 1.09·17-s + 1.44·18-s + 1.52·19-s + 1.33·20-s − 0.945·21-s + 1.25·22-s − 0.870·23-s − 2.67·24-s − 0.607·25-s + 0.249·27-s + 1.49·28-s − 0.525·29-s − 1.49·30-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.741751828\)
\(L(\frac12)\) \(\approx\) \(3.741751828\)
\(L(\frac{5}{2})\) 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 \( 1 - 5 T + p^{3} T^{2} \)
3 \( 1 + 7 T + p^{3} T^{2} \)
5 \( 1 - 7 T + p^{3} T^{2} \)
7 \( 1 - 13 T + p^{3} T^{2} \)
11 \( 1 - 26 T + p^{3} T^{2} \)
17 \( 1 - 77 T + p^{3} T^{2} \)
19 \( 1 - 126 T + p^{3} T^{2} \)
23 \( 1 + 96 T + p^{3} T^{2} \)
29 \( 1 + 82 T + p^{3} T^{2} \)
31 \( 1 + 196 T + p^{3} T^{2} \)
37 \( 1 - 131 T + p^{3} T^{2} \)
41 \( 1 + 336 T + p^{3} T^{2} \)
43 \( 1 + 201 T + p^{3} T^{2} \)
47 \( 1 - 105 T + p^{3} T^{2} \)
53 \( 1 + 432 T + p^{3} T^{2} \)
59 \( 1 - 294 T + p^{3} T^{2} \)
61 \( 1 + 56 T + p^{3} T^{2} \)
67 \( 1 + 478 T + p^{3} T^{2} \)
71 \( 1 + 9 T + p^{3} T^{2} \)
73 \( 1 + 98 T + p^{3} T^{2} \)
79 \( 1 - 1304 T + p^{3} T^{2} \)
83 \( 1 - 308 T + p^{3} T^{2} \)
89 \( 1 - 1190 T + p^{3} T^{2} \)
97 \( 1 + 70 T + p^{3} T^{2} \)
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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

−12.06982900588716248438967969865, −11.80434401370175741148290795958, −10.90742058065693076721969271373, −9.716506956626598909344307784643, −7.58851824789922030946101308103, −6.37084450086936827568846446539, −5.58744463371738230345339955750, −4.97680026459199100254139694471, −3.57335445455789660377818728774, −1.61421828074687945911669584549, 1.61421828074687945911669584549, 3.57335445455789660377818728774, 4.97680026459199100254139694471, 5.58744463371738230345339955750, 6.37084450086936827568846446539, 7.58851824789922030946101308103, 9.716506956626598909344307784643, 10.90742058065693076721969271373, 11.80434401370175741148290795958, 12.06982900588716248438967969865

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