Properties

Label 2-297-1.1-c1-0-0
Degree $2$
Conductor $297$
Sign $1$
Analytic cond. $2.37155$
Root an. cond. $1.53998$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.08·2-s + 2.35·4-s − 1.35·5-s − 0.0861·7-s − 0.734·8-s + 2.82·10-s − 11-s + 0.648·13-s + 0.179·14-s − 3.17·16-s + 3.43·17-s + 4.82·19-s − 3.17·20-s + 2.08·22-s + 5.82·23-s − 3.17·25-s − 1.35·26-s − 0.202·28-s + 4.79·29-s + 0.648·31-s + 8.08·32-s − 7.17·34-s + 0.116·35-s + 11.4·37-s − 10.0·38-s + 0.992·40-s + 8.96·41-s + ⋯
L(s)  = 1  − 1.47·2-s + 1.17·4-s − 0.604·5-s − 0.0325·7-s − 0.259·8-s + 0.891·10-s − 0.301·11-s + 0.179·13-s + 0.0480·14-s − 0.793·16-s + 0.833·17-s + 1.10·19-s − 0.710·20-s + 0.444·22-s + 1.21·23-s − 0.634·25-s − 0.265·26-s − 0.0382·28-s + 0.889·29-s + 0.116·31-s + 1.42·32-s − 1.23·34-s + 0.0196·35-s + 1.87·37-s − 1.63·38-s + 0.156·40-s + 1.39·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $1$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5697023630\)
\(L(\frac12)\) \(\approx\) \(0.5697023630\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + T \)
good2 \( 1 + 2.08T + 2T^{2} \)
5 \( 1 + 1.35T + 5T^{2} \)
7 \( 1 + 0.0861T + 7T^{2} \)
13 \( 1 - 0.648T + 13T^{2} \)
17 \( 1 - 3.43T + 17T^{2} \)
19 \( 1 - 4.82T + 19T^{2} \)
23 \( 1 - 5.82T + 23T^{2} \)
29 \( 1 - 4.79T + 29T^{2} \)
31 \( 1 - 0.648T + 31T^{2} \)
37 \( 1 - 11.4T + 37T^{2} \)
41 \( 1 - 8.96T + 41T^{2} \)
43 \( 1 - 9.61T + 43T^{2} \)
47 \( 1 - 3.87T + 47T^{2} \)
53 \( 1 + 7.46T + 53T^{2} \)
59 \( 1 + 6.69T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + 4.99T + 67T^{2} \)
71 \( 1 + 7.46T + 71T^{2} \)
73 \( 1 + 4.87T + 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 - 1.94T + 83T^{2} \)
89 \( 1 + 5.04T + 89T^{2} \)
97 \( 1 - 15.5T + 97T^{2} \)
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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

−11.40298043576743741361810628562, −10.71955620622961990041550383491, −9.685090731375757474944210711941, −9.038076328950401776070553024869, −7.75928342783066699332770290749, −7.58164261758924417529885042586, −6.09578860908516564921402145127, −4.56950444803272845120769088935, −2.93181297524400576082358356124, −1.00944501760490582867319683839, 1.00944501760490582867319683839, 2.93181297524400576082358356124, 4.56950444803272845120769088935, 6.09578860908516564921402145127, 7.58164261758924417529885042586, 7.75928342783066699332770290749, 9.038076328950401776070553024869, 9.685090731375757474944210711941, 10.71955620622961990041550383491, 11.40298043576743741361810628562

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