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Properties

Label 2-297-9.4-c1-0-6
Degree $2$
Conductor $297$
Sign $-0.939 + 0.342i$
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  + (−0.939 − 1.62i)2-s + (−0.766 + 1.32i)4-s + (1.43 − 2.49i)5-s + (0.326 + 0.565i)7-s − 0.879·8-s − 5.41·10-s + (−0.5 − 0.866i)11-s + (3.37 − 5.85i)13-s + (0.613 − 1.06i)14-s + (2.35 + 4.08i)16-s − 0.184·17-s − 5.22·19-s + (2.20 + 3.82i)20-s + (−0.939 + 1.62i)22-s + (−1.59 + 2.75i)23-s + ⋯
L(s)  = 1  + (−0.664 − 1.15i)2-s + (−0.383 + 0.663i)4-s + (0.643 − 1.11i)5-s + (0.123 + 0.213i)7-s − 0.310·8-s − 1.71·10-s + (−0.150 − 0.261i)11-s + (0.937 − 1.62i)13-s + (0.163 − 0.283i)14-s + (0.589 + 1.02i)16-s − 0.0448·17-s − 1.19·19-s + (0.493 + 0.854i)20-s + (−0.200 + 0.347i)22-s + (−0.332 + 0.575i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.162391 - 0.920966i\)
\(L(\frac12)\) \(\approx\) \(0.162391 - 0.920966i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.939 + 1.62i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.43 + 2.49i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.326 - 0.565i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (-3.37 + 5.85i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.184T + 17T^{2} \)
19 \( 1 + 5.22T + 19T^{2} \)
23 \( 1 + (1.59 - 2.75i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.01 + 3.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.553 - 0.957i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.106T + 37T^{2} \)
41 \( 1 + (2.80 - 4.86i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.92 + 3.34i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.00 - 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 + (-5.27 + 9.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.67 - 6.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.90 + 10.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.47T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 + (-0.733 - 1.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.520 + 0.902i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.01T + 89T^{2} \)
97 \( 1 + (-2.86 - 4.97i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.15163333380977222181912900135, −10.42647146572784329545542944206, −9.573707659301433004451754353606, −8.661215908162822105077658541302, −8.122907256293427768156238648713, −6.11004028884001231456362969889, −5.32064532355896605075344769924, −3.67436588405866149105852126516, −2.22081221963308618881918913969, −0.884530217831679671929488702766, 2.24683531682321304980551133672, 3.98648796847516815450578457915, 5.70144495798354710881880417766, 6.70535181082041685499618110259, 6.97548671456027712794225657824, 8.372300765466276839193214309410, 9.103120815496979919447757794946, 10.17239154420141317267422700749, 10.96180165205733018514587104901, 12.05163482902305795797138903105

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