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Properties

Label 2-297-11.5-c1-0-15
Degree $2$
Conductor $297$
Sign $-0.897 - 0.440i$
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  + (−0.565 − 1.74i)2-s + (−1.09 + 0.796i)4-s + (0.517 − 1.59i)5-s + (−3.34 + 2.43i)7-s + (−0.956 − 0.695i)8-s − 3.06·10-s + (−3.30 − 0.318i)11-s + (−1.63 − 5.03i)13-s + (6.12 + 4.45i)14-s + (−1.50 + 4.63i)16-s + (1.79 − 5.51i)17-s + (−0.756 − 0.549i)19-s + (0.700 + 2.15i)20-s + (1.31 + 5.93i)22-s − 1.01·23-s + ⋯
L(s)  = 1  + (−0.400 − 1.23i)2-s + (−0.547 + 0.398i)4-s + (0.231 − 0.711i)5-s + (−1.26 + 0.918i)7-s + (−0.338 − 0.245i)8-s − 0.969·10-s + (−0.995 − 0.0959i)11-s + (−0.453 − 1.39i)13-s + (1.63 + 1.18i)14-s + (−0.376 + 1.15i)16-s + (0.434 − 1.33i)17-s + (−0.173 − 0.126i)19-s + (0.156 + 0.481i)20-s + (0.280 + 1.26i)22-s − 0.210·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-0.897 - 0.440i$
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: \((2,\ 297,\ (\ :1/2),\ -0.897 - 0.440i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.131594 + 0.567072i\)
\(L(\frac12)\) \(\approx\) \(0.131594 + 0.567072i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.30 + 0.318i)T \)
good2 \( 1 + (0.565 + 1.74i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-0.517 + 1.59i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (3.34 - 2.43i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.63 + 5.03i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.79 + 5.51i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.756 + 0.549i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 1.01T + 23T^{2} \)
29 \( 1 + (3.38 - 2.45i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.398 - 1.22i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-8.02 + 5.82i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.12 - 0.818i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 7.55T + 43T^{2} \)
47 \( 1 + (4.77 + 3.46i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.303 - 0.933i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.59 - 2.61i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.42 + 10.5i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 2.75T + 67T^{2} \)
71 \( 1 + (-2.76 + 8.51i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (9.56 - 6.94i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.44 + 10.5i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.39 + 13.5i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 7.12T + 89T^{2} \)
97 \( 1 + (1.08 + 3.33i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.17219959826427853494912982749, −10.18664419819298061829844961582, −9.519008642741851143715273433949, −8.865344494063167839347483260811, −7.56676632291642540589187766051, −6.00281938173767840335060117932, −5.14232343959362178211375986840, −3.20343744677046179252703711658, −2.52015051312024230663142440983, −0.45622704339792095926250246404, 2.70018871922555920616806925453, 4.18379214143735977919576305089, 5.87911873583276280102784152824, 6.58221272774580115262463799798, 7.28945458675233556379687728923, 8.193737961903947922558732523484, 9.510307998664373736487389720577, 10.11227566829738171370121861095, 11.13568432919243792024646591401, 12.47118732011966767866057235698

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