Properties

Degree $2$
Conductor $2169$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·2-s + 1.01·4-s + 2.63·5-s − 2.01·7-s + 1.70·8-s − 4.57·10-s + 3.39·11-s + 5.63·13-s + 3.49·14-s − 4.99·16-s − 0.866·17-s + 2.46·19-s + 2.67·20-s − 5.89·22-s + 6.37·23-s + 1.92·25-s − 9.79·26-s − 2.04·28-s + 4.52·29-s − 3.51·31-s + 5.26·32-s + 1.50·34-s − 5.29·35-s − 5.19·37-s − 4.28·38-s + 4.49·40-s − 1.35·41-s + ⋯
L(s)  = 1  − 1.22·2-s + 0.508·4-s + 1.17·5-s − 0.759·7-s + 0.603·8-s − 1.44·10-s + 1.02·11-s + 1.56·13-s + 0.933·14-s − 1.24·16-s − 0.210·17-s + 0.565·19-s + 0.598·20-s − 1.25·22-s + 1.33·23-s + 0.385·25-s − 1.91·26-s − 0.386·28-s + 0.839·29-s − 0.631·31-s + 0.931·32-s + 0.258·34-s − 0.894·35-s − 0.853·37-s − 0.694·38-s + 0.710·40-s − 0.211·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2169\)    =    \(3^{2} \cdot 241\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{2169} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2169,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.240303174\)
\(L(\frac12)\) \(\approx\) \(1.240303174\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
241 \( 1 + T \)
good2 \( 1 + 1.73T + 2T^{2} \)
5 \( 1 - 2.63T + 5T^{2} \)
7 \( 1 + 2.01T + 7T^{2} \)
11 \( 1 - 3.39T + 11T^{2} \)
13 \( 1 - 5.63T + 13T^{2} \)
17 \( 1 + 0.866T + 17T^{2} \)
19 \( 1 - 2.46T + 19T^{2} \)
23 \( 1 - 6.37T + 23T^{2} \)
29 \( 1 - 4.52T + 29T^{2} \)
31 \( 1 + 3.51T + 31T^{2} \)
37 \( 1 + 5.19T + 37T^{2} \)
41 \( 1 + 1.35T + 41T^{2} \)
43 \( 1 - 8.49T + 43T^{2} \)
47 \( 1 - 9.44T + 47T^{2} \)
53 \( 1 + 9.71T + 53T^{2} \)
59 \( 1 + 6.03T + 59T^{2} \)
61 \( 1 - 4.45T + 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 - 3.01T + 71T^{2} \)
73 \( 1 + 0.255T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 - 16.8T + 83T^{2} \)
89 \( 1 - 17.4T + 89T^{2} \)
97 \( 1 - 0.273T + 97T^{2} \)
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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

−9.215346848300112174187822251490, −8.700361676308640167163511130896, −7.63369407186685769921485397581, −6.67402879730091914292647849019, −6.26945555834336181035110482977, −5.28857168404428725481032051845, −4.08326628800448549368667722083, −3.04668305644489591501918654679, −1.71914014257974484895952677401, −0.966451702440892308446168591401, 0.966451702440892308446168591401, 1.71914014257974484895952677401, 3.04668305644489591501918654679, 4.08326628800448549368667722083, 5.28857168404428725481032051845, 6.26945555834336181035110482977, 6.67402879730091914292647849019, 7.63369407186685769921485397581, 8.700361676308640167163511130896, 9.215346848300112174187822251490

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