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Properties

Label 2-297-11.4-c1-0-15
Degree $2$
Conductor $297$
Sign $-0.915 - 0.402i$
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  + (−0.863 − 0.627i)2-s + (−0.266 − 0.819i)4-s + (−0.993 + 0.721i)5-s + (−0.473 − 1.45i)7-s + (−0.943 + 2.90i)8-s + 1.30·10-s + (−3.00 − 1.40i)11-s + (−1.89 − 1.37i)13-s + (−0.505 + 1.55i)14-s + (1.24 − 0.900i)16-s + (−2.87 + 2.08i)17-s + (−0.884 + 2.72i)19-s + (0.856 + 0.622i)20-s + (1.70 + 3.09i)22-s − 2.29·23-s + ⋯
L(s)  = 1  + (−0.610 − 0.443i)2-s + (−0.133 − 0.409i)4-s + (−0.444 + 0.322i)5-s + (−0.179 − 0.550i)7-s + (−0.333 + 1.02i)8-s + 0.414·10-s + (−0.905 − 0.424i)11-s + (−0.525 − 0.381i)13-s + (−0.135 + 0.415i)14-s + (0.310 − 0.225i)16-s + (−0.696 + 0.506i)17-s + (−0.202 + 0.624i)19-s + (0.191 + 0.139i)20-s + (0.364 + 0.660i)22-s − 0.478·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.402i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-0.915 - 0.402i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ -0.915 - 0.402i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0343438 + 0.163395i\)
\(L(\frac12)\) \(\approx\) \(0.0343438 + 0.163395i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.00 + 1.40i)T \)
good2 \( 1 + (0.863 + 0.627i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (0.993 - 0.721i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.473 + 1.45i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.89 + 1.37i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.87 - 2.08i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.884 - 2.72i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.29T + 23T^{2} \)
29 \( 1 + (2.56 + 7.88i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.91 + 2.84i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.24 - 3.82i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.49 - 4.60i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 7.52T + 43T^{2} \)
47 \( 1 + (-2.82 + 8.70i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.00 - 3.63i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.36 + 10.3i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.496 + 0.360i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 8.65T + 67T^{2} \)
71 \( 1 + (-12.6 + 9.18i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.68 + 14.4i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.57 - 1.14i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.31 - 3.13i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 5.84T + 89T^{2} \)
97 \( 1 + (-6.88 - 4.99i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.01785488061966085198825763108, −10.30992993084841102244650156866, −9.612288591408481281068213843944, −8.337049816485317838465616202152, −7.65744000420305026361764695214, −6.25973518668195910053589317502, −5.15532487909200089231144101595, −3.70880228987770723634052324784, −2.17848557313295743388748027779, −0.14107557477467729452769360210, 2.58842987220199569296241443061, 4.12326549315683941562656031929, 5.27978323363656945273410871358, 6.81365534189727916096868867584, 7.51223564356333485912094512236, 8.574328336225043351652546671242, 9.161330873096570337317671737964, 10.19287018484410279695984229918, 11.41536759079585832460009151459, 12.46135186472887520806645283054

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