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Properties

Label 2-2e9-512.269-c1-0-35
Degree $2$
Conductor $512$
Sign $0.590 - 0.806i$
Analytic cond. $4.08834$
Root an. cond. $2.02196$
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  + (−0.219 + 1.39i)2-s + (3.00 − 1.51i)3-s + (−1.90 − 0.614i)4-s + (1.57 + 3.56i)5-s + (1.45 + 4.52i)6-s + (−2.44 − 1.46i)7-s + (1.27 − 2.52i)8-s + (4.94 − 6.66i)9-s + (−5.32 + 1.42i)10-s + (2.22 + 1.73i)11-s + (−6.64 + 1.03i)12-s + (−0.222 + 0.211i)13-s + (2.58 − 3.09i)14-s + (10.1 + 8.30i)15-s + (3.24 + 2.33i)16-s + (3.38 + 4.12i)17-s + ⋯
L(s)  = 1  + (−0.155 + 0.987i)2-s + (1.73 − 0.872i)3-s + (−0.951 − 0.307i)4-s + (0.706 + 1.59i)5-s + (0.592 + 1.84i)6-s + (−0.923 − 0.553i)7-s + (0.451 − 0.892i)8-s + (1.64 − 2.22i)9-s + (−1.68 + 0.449i)10-s + (0.672 + 0.524i)11-s + (−1.91 + 0.297i)12-s + (−0.0616 + 0.0587i)13-s + (0.690 − 0.826i)14-s + (2.61 + 2.14i)15-s + (0.811 + 0.584i)16-s + (0.821 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.590 - 0.806i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 0.590 - 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01242 + 1.02071i\)
\(L(\frac12)\) \(\approx\) \(2.01242 + 1.02071i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.219 - 1.39i)T \)
good3 \( 1 + (-3.00 + 1.51i)T + (1.78 - 2.40i)T^{2} \)
5 \( 1 + (-1.57 - 3.56i)T + (-3.35 + 3.70i)T^{2} \)
7 \( 1 + (2.44 + 1.46i)T + (3.29 + 6.17i)T^{2} \)
11 \( 1 + (-2.22 - 1.73i)T + (2.67 + 10.6i)T^{2} \)
13 \( 1 + (0.222 - 0.211i)T + (0.637 - 12.9i)T^{2} \)
17 \( 1 + (-3.38 - 4.12i)T + (-3.31 + 16.6i)T^{2} \)
19 \( 1 + (0.400 + 2.30i)T + (-17.8 + 6.40i)T^{2} \)
23 \( 1 + (3.85 + 1.82i)T + (14.5 + 17.7i)T^{2} \)
29 \( 1 + (-5.21 - 2.95i)T + (14.9 + 24.8i)T^{2} \)
31 \( 1 + (5.39 - 1.07i)T + (28.6 - 11.8i)T^{2} \)
37 \( 1 + (4.88 + 3.09i)T + (15.8 + 33.4i)T^{2} \)
41 \( 1 + (-0.977 - 1.07i)T + (-4.01 + 40.8i)T^{2} \)
43 \( 1 + (0.965 + 2.92i)T + (-34.5 + 25.6i)T^{2} \)
47 \( 1 + (4.13 + 2.21i)T + (26.1 + 39.0i)T^{2} \)
53 \( 1 + (0.430 - 0.244i)T + (27.2 - 45.4i)T^{2} \)
59 \( 1 + (-3.55 + 3.73i)T + (-2.89 - 58.9i)T^{2} \)
61 \( 1 + (-0.0285 - 0.387i)T + (-60.3 + 8.95i)T^{2} \)
67 \( 1 + (9.90 + 8.54i)T + (9.83 + 66.2i)T^{2} \)
71 \( 1 + (-0.752 - 0.111i)T + (67.9 + 20.6i)T^{2} \)
73 \( 1 + (0.363 - 0.217i)T + (34.4 - 64.3i)T^{2} \)
79 \( 1 + (-5.17 - 1.57i)T + (65.6 + 43.8i)T^{2} \)
83 \( 1 + (7.19 - 4.55i)T + (35.4 - 75.0i)T^{2} \)
89 \( 1 + (16.2 - 7.66i)T + (56.4 - 68.7i)T^{2} \)
97 \( 1 + (-4.42 - 2.95i)T + (37.1 + 89.6i)T^{2} \)
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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

−10.40359727381213685990671524721, −9.895223678408825531158029290585, −9.159199446503124541638712551487, −8.158834812941129337947987929215, −7.13900002048240566468936990346, −6.84618172539999444504588702844, −6.13049447487073367864457097015, −3.87505294130631257099753866571, −3.19279451201114981670161244947, −1.79713698608716775321494086678, 1.55460231752528675767368527616, 2.77188807422047580878282251150, 3.70322795553259293150817987577, 4.66835152787083486639224077320, 5.66381048855784320925633934898, 7.81870419155386790794842742486, 8.666708575577139363684998450018, 9.091151722724857890448596035474, 9.773375264295587486735130242493, 10.11727930652683533301106225310

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