Properties

Label 2-13e2-1.1-c3-0-14
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  − 5.42·2-s + 1.67·3-s + 21.4·4-s − 7.70·5-s − 9.09·6-s + 15.0·7-s − 73.1·8-s − 24.1·9-s + 41.8·10-s + 2.51·11-s + 35.9·12-s − 81.5·14-s − 12.9·15-s + 225.·16-s − 2.06·17-s + 131.·18-s + 94.4·19-s − 165.·20-s + 25.1·21-s − 13.6·22-s + 35.8·23-s − 122.·24-s − 65.5·25-s − 85.7·27-s + 322.·28-s − 140.·29-s + 70.0·30-s + ⋯
L(s)  = 1  − 1.91·2-s + 0.322·3-s + 2.68·4-s − 0.689·5-s − 0.618·6-s + 0.811·7-s − 3.23·8-s − 0.896·9-s + 1.32·10-s + 0.0688·11-s + 0.865·12-s − 1.55·14-s − 0.222·15-s + 3.51·16-s − 0.0294·17-s + 1.71·18-s + 1.14·19-s − 1.85·20-s + 0.261·21-s − 0.132·22-s + 0.325·23-s − 1.04·24-s − 0.524·25-s − 0.611·27-s + 2.17·28-s − 0.897·29-s + 0.426·30-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 \( 1 + 5.42T + 8T^{2} \)
3 \( 1 - 1.67T + 27T^{2} \)
5 \( 1 + 7.70T + 125T^{2} \)
7 \( 1 - 15.0T + 343T^{2} \)
11 \( 1 - 2.51T + 1.33e3T^{2} \)
17 \( 1 + 2.06T + 4.91e3T^{2} \)
19 \( 1 - 94.4T + 6.85e3T^{2} \)
23 \( 1 - 35.8T + 1.21e4T^{2} \)
29 \( 1 + 140.T + 2.43e4T^{2} \)
31 \( 1 + 264.T + 2.97e4T^{2} \)
37 \( 1 + 256.T + 5.06e4T^{2} \)
41 \( 1 + 394.T + 6.89e4T^{2} \)
43 \( 1 - 256.T + 7.95e4T^{2} \)
47 \( 1 - 415.T + 1.03e5T^{2} \)
53 \( 1 + 504.T + 1.48e5T^{2} \)
59 \( 1 + 120.T + 2.05e5T^{2} \)
61 \( 1 + 752.T + 2.26e5T^{2} \)
67 \( 1 + 211.T + 3.00e5T^{2} \)
71 \( 1 - 410.T + 3.57e5T^{2} \)
73 \( 1 + 17.4T + 3.89e5T^{2} \)
79 \( 1 + 174.T + 4.93e5T^{2} \)
83 \( 1 + 963.T + 5.71e5T^{2} \)
89 \( 1 + 477.T + 7.04e5T^{2} \)
97 \( 1 + 1.04e3T + 9.12e5T^{2} \)
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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

−11.39304712439483342835664318271, −10.83925152268746980642226213190, −9.493599203780949666903187232567, −8.733871421012563587040804295836, −7.86817478077990652474811639077, −7.23098226286307010123434178703, −5.62643886739790634714486578996, −3.27262693911283379962573314566, −1.72006127197663965269202435722, 0, 1.72006127197663965269202435722, 3.27262693911283379962573314566, 5.62643886739790634714486578996, 7.23098226286307010123434178703, 7.86817478077990652474811639077, 8.733871421012563587040804295836, 9.493599203780949666903187232567, 10.83925152268746980642226213190, 11.39304712439483342835664318271

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