Properties

Label 2-13e2-1.1-c3-0-8
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  − 0.390·2-s + 3.60·3-s − 7.84·4-s − 7.52·5-s − 1.40·6-s + 19.5·7-s + 6.18·8-s − 13.9·9-s + 2.93·10-s + 45.8·11-s − 28.3·12-s − 7.62·14-s − 27.1·15-s + 60.3·16-s + 86.5·17-s + 5.45·18-s + 148.·19-s + 59.0·20-s + 70.5·21-s − 17.8·22-s − 91.5·23-s + 22.3·24-s − 68.4·25-s − 147.·27-s − 153.·28-s + 258.·29-s + 10.5·30-s + ⋯
L(s)  = 1  − 0.137·2-s + 0.694·3-s − 0.980·4-s − 0.672·5-s − 0.0958·6-s + 1.05·7-s + 0.273·8-s − 0.517·9-s + 0.0927·10-s + 1.25·11-s − 0.681·12-s − 0.145·14-s − 0.467·15-s + 0.943·16-s + 1.23·17-s + 0.0713·18-s + 1.79·19-s + 0.659·20-s + 0.733·21-s − 0.173·22-s − 0.829·23-s + 0.189·24-s − 0.547·25-s − 1.05·27-s − 1.03·28-s + 1.65·29-s + 0.0644·30-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

\(L(2)\) \(\approx\) \(1.663186260\)
\(L(\frac12)\) \(\approx\) \(1.663186260\)
\(L(\frac{5}{2})\) 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 \( 1 + 0.390T + 8T^{2} \)
3 \( 1 - 3.60T + 27T^{2} \)
5 \( 1 + 7.52T + 125T^{2} \)
7 \( 1 - 19.5T + 343T^{2} \)
11 \( 1 - 45.8T + 1.33e3T^{2} \)
17 \( 1 - 86.5T + 4.91e3T^{2} \)
19 \( 1 - 148.T + 6.85e3T^{2} \)
23 \( 1 + 91.5T + 1.21e4T^{2} \)
29 \( 1 - 258.T + 2.43e4T^{2} \)
31 \( 1 - 31.2T + 2.97e4T^{2} \)
37 \( 1 - 148.T + 5.06e4T^{2} \)
41 \( 1 - 95.9T + 6.89e4T^{2} \)
43 \( 1 + 80.9T + 7.95e4T^{2} \)
47 \( 1 - 94.3T + 1.03e5T^{2} \)
53 \( 1 - 493.T + 1.48e5T^{2} \)
59 \( 1 + 575.T + 2.05e5T^{2} \)
61 \( 1 + 40.2T + 2.26e5T^{2} \)
67 \( 1 - 601.T + 3.00e5T^{2} \)
71 \( 1 + 518.T + 3.57e5T^{2} \)
73 \( 1 + 1.05e3T + 3.89e5T^{2} \)
79 \( 1 + 320.T + 4.93e5T^{2} \)
83 \( 1 + 32.4T + 5.71e5T^{2} \)
89 \( 1 - 450.T + 7.04e5T^{2} \)
97 \( 1 - 231.T + 9.12e5T^{2} \)
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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

−12.01712164814778279426859421254, −11.66297656465704198516348012154, −10.04213958262747740338664794806, −9.121077306803018730179518342169, −8.184761353818233308126072410603, −7.63984177622862917323455221424, −5.66234074927463592786155070105, −4.38274930129402736804609056663, −3.33218043576476035395280837059, −1.12804639511102488251946898927, 1.12804639511102488251946898927, 3.33218043576476035395280837059, 4.38274930129402736804609056663, 5.66234074927463592786155070105, 7.63984177622862917323455221424, 8.184761353818233308126072410603, 9.121077306803018730179518342169, 10.04213958262747740338664794806, 11.66297656465704198516348012154, 12.01712164814778279426859421254

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