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Properties

Label 2-297-11.5-c1-0-7
Degree $2$
Conductor $297$
Sign $0.833 + 0.553i$
Analytic cond. $2.37155$
Root an. cond. $1.53998$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0821 − 0.252i)2-s + (1.56 − 1.13i)4-s + (−0.340 + 1.04i)5-s + (1.24 − 0.907i)7-s + (−0.845 − 0.614i)8-s + 0.292·10-s + (3.23 + 0.743i)11-s + (−0.254 − 0.782i)13-s + (−0.332 − 0.241i)14-s + (1.10 − 3.40i)16-s + (0.295 − 0.907i)17-s + (0.281 + 0.204i)19-s + (0.656 + 2.01i)20-s + (−0.0775 − 0.878i)22-s − 6.09·23-s + ⋯
L(s)  = 1  + (−0.0581 − 0.178i)2-s + (0.780 − 0.567i)4-s + (−0.152 + 0.468i)5-s + (0.472 − 0.342i)7-s + (−0.298 − 0.217i)8-s + 0.0925·10-s + (0.974 + 0.224i)11-s + (−0.0705 − 0.216i)13-s + (−0.0887 − 0.0644i)14-s + (0.276 − 0.851i)16-s + (0.0715 − 0.220i)17-s + (0.0645 + 0.0469i)19-s + (0.146 + 0.451i)20-s + (−0.0165 − 0.187i)22-s − 1.27·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.833 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.833 + 0.553i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ 0.833 + 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46845 - 0.443015i\)
\(L(\frac12)\) \(\approx\) \(1.46845 - 0.443015i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.23 - 0.743i)T \)
good2 \( 1 + (0.0821 + 0.252i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (0.340 - 1.04i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.24 + 0.907i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.254 + 0.782i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.295 + 0.907i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.281 - 0.204i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 6.09T + 23T^{2} \)
29 \( 1 + (-5.70 + 4.14i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.07 - 3.30i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.15 - 0.837i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (7.57 + 5.50i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 6.57T + 43T^{2} \)
47 \( 1 + (-5.14 - 3.73i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.39 - 7.36i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.52 - 5.46i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.38 - 10.4i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 9.67T + 67T^{2} \)
71 \( 1 + (-0.246 + 0.757i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (11.0 - 8.01i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.884 + 2.72i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.63 + 14.2i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 7.49T + 89T^{2} \)
97 \( 1 + (-1.81 - 5.59i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.77530040594043577965257944167, −10.62973588842647366521512903623, −10.13278986189772531435535369266, −8.923506174317850566182997917425, −7.61717754276865373366542614756, −6.79909544112798742595851864566, −5.85865560429881198159475825324, −4.43251369364659192384293697769, −2.98526010623206743646217265811, −1.47333071020166870966305467735, 1.81590906350740473458828828588, 3.39734182907849640526043321920, 4.68827456721515930877267445465, 6.09401752257620502633567471343, 6.93466948946851473601314318701, 8.203640480656603465602958240235, 8.650081053128382882426065720673, 9.973344300042040945202459650306, 11.15863081762001971786192629407, 11.95983104334351049169698134523

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