Properties

Label 2-13e2-1.1-c3-0-10
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  − 2.22·2-s − 9.74·3-s − 3.03·4-s + 8.20·5-s + 21.7·6-s − 8.35·7-s + 24.5·8-s + 68.0·9-s − 18.2·10-s + 9.69·11-s + 29.5·12-s + 18.6·14-s − 80.0·15-s − 30.4·16-s + 44.6·17-s − 151.·18-s − 87.7·19-s − 24.9·20-s + 81.4·21-s − 21.6·22-s + 107.·23-s − 239.·24-s − 57.6·25-s − 400.·27-s + 25.3·28-s − 14.0·29-s + 178.·30-s + ⋯
L(s)  = 1  − 0.787·2-s − 1.87·3-s − 0.379·4-s + 0.734·5-s + 1.47·6-s − 0.451·7-s + 1.08·8-s + 2.51·9-s − 0.578·10-s + 0.265·11-s + 0.712·12-s + 0.355·14-s − 1.37·15-s − 0.476·16-s + 0.636·17-s − 1.98·18-s − 1.05·19-s − 0.278·20-s + 0.846·21-s − 0.209·22-s + 0.970·23-s − 2.03·24-s − 0.461·25-s − 2.85·27-s + 0.171·28-s − 0.0899·29-s + 1.08·30-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 \( 1 + 2.22T + 8T^{2} \)
3 \( 1 + 9.74T + 27T^{2} \)
5 \( 1 - 8.20T + 125T^{2} \)
7 \( 1 + 8.35T + 343T^{2} \)
11 \( 1 - 9.69T + 1.33e3T^{2} \)
17 \( 1 - 44.6T + 4.91e3T^{2} \)
19 \( 1 + 87.7T + 6.85e3T^{2} \)
23 \( 1 - 107.T + 1.21e4T^{2} \)
29 \( 1 + 14.0T + 2.43e4T^{2} \)
31 \( 1 + 171.T + 2.97e4T^{2} \)
37 \( 1 - 413.T + 5.06e4T^{2} \)
41 \( 1 + 258.T + 6.89e4T^{2} \)
43 \( 1 - 61.0T + 7.95e4T^{2} \)
47 \( 1 - 68.7T + 1.03e5T^{2} \)
53 \( 1 - 328.T + 1.48e5T^{2} \)
59 \( 1 + 147.T + 2.05e5T^{2} \)
61 \( 1 + 97.8T + 2.26e5T^{2} \)
67 \( 1 + 677.T + 3.00e5T^{2} \)
71 \( 1 + 786.T + 3.57e5T^{2} \)
73 \( 1 + 997.T + 3.89e5T^{2} \)
79 \( 1 - 383.T + 4.93e5T^{2} \)
83 \( 1 + 519.T + 5.71e5T^{2} \)
89 \( 1 + 683.T + 7.04e5T^{2} \)
97 \( 1 - 347.T + 9.12e5T^{2} \)
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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

−11.60064990277485616077506995334, −10.62225450047243612479210665615, −9.988854325161588559371948879754, −9.109846919713455114265875701045, −7.46724101020689112777156737474, −6.35686169349934092412217739804, −5.46446153733568983887636407065, −4.31230412326737361890709507409, −1.37696297196305850582859016819, 0, 1.37696297196305850582859016819, 4.31230412326737361890709507409, 5.46446153733568983887636407065, 6.35686169349934092412217739804, 7.46724101020689112777156737474, 9.109846919713455114265875701045, 9.988854325161588559371948879754, 10.62225450047243612479210665615, 11.60064990277485616077506995334

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