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Properties

Label 2-297-99.43-c0-0-0
Degree $2$
Conductor $297$
Sign $0.342 + 0.939i$
Analytic cond. $0.148222$
Root an. cond. $0.384996$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)11-s + (−0.499 + 0.866i)16-s + (−0.499 + 0.866i)20-s + (1 + 1.73i)23-s + (0.5 + 0.866i)31-s − 37-s − 0.999·44-s + (−0.5 + 0.866i)47-s + (−0.5 − 0.866i)49-s + 53-s − 0.999·55-s + (−0.5 − 0.866i)59-s + 0.999·64-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)11-s + (−0.499 + 0.866i)16-s + (−0.499 + 0.866i)20-s + (1 + 1.73i)23-s + (0.5 + 0.866i)31-s − 37-s − 0.999·44-s + (−0.5 + 0.866i)47-s + (−0.5 − 0.866i)49-s + 53-s − 0.999·55-s + (−0.5 − 0.866i)59-s + 0.999·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(0.148222\)
Root analytic conductor: \(0.384996\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :0),\ 0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6583042175\)
\(L(\frac12)\) \(\approx\) \(0.6583042175\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−11.71818531332266187221612733754, −10.94443428055690846751617338561, −9.796016467305744821634256055648, −8.946546214429717118093823632696, −8.279897436479740655662250885709, −6.84159616035865893324531911570, −5.61229297238830216896924244978, −4.80518956208838619905369878599, −3.56493651180833204370871519282, −1.23510019716345420483302233806, 2.67917152496581090293312586167, 3.84413042026171845673964585907, 4.83107582838988357916337143368, 6.62769648388861134426301840169, 7.29776356599962325506388115497, 8.320782544654494967207437478526, 9.255379467038931507014275600433, 10.35009411927364674340876402387, 11.34076781514149823181670269102, 12.20025696734899366629885966017

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