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Properties

Label 2-2e9-128.101-c1-0-6
Degree $2$
Conductor $512$
Sign $0.561 - 0.827i$
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  + (0.135 − 0.0724i)3-s + (2.12 + 2.59i)5-s + (1.11 − 0.742i)7-s + (−1.65 + 2.47i)9-s + (0.181 − 0.0550i)11-s + (0.374 + 0.307i)13-s + (0.476 + 0.197i)15-s + (1.79 − 0.742i)17-s + (0.487 + 4.94i)19-s + (0.0968 − 0.181i)21-s + (−5.40 + 1.07i)23-s + (−1.22 + 6.14i)25-s + (−0.0900 + 0.914i)27-s + (2.45 − 8.10i)29-s + (−3.30 − 3.30i)31-s + ⋯
L(s)  = 1  + (0.0783 − 0.0418i)3-s + (0.952 + 1.16i)5-s + (0.419 − 0.280i)7-s + (−0.551 + 0.824i)9-s + (0.0547 − 0.0166i)11-s + (0.103 + 0.0851i)13-s + (0.123 + 0.0510i)15-s + (0.434 − 0.180i)17-s + (0.111 + 1.13i)19-s + (0.0211 − 0.0395i)21-s + (−1.12 + 0.224i)23-s + (−0.244 + 1.22i)25-s + (−0.0173 + 0.176i)27-s + (0.456 − 1.50i)29-s + (−0.593 − 0.593i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.561 - 0.827i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 0.561 - 0.827i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50171 + 0.795702i\)
\(L(\frac12)\) \(\approx\) \(1.50171 + 0.795702i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-0.135 + 0.0724i)T + (1.66 - 2.49i)T^{2} \)
5 \( 1 + (-2.12 - 2.59i)T + (-0.975 + 4.90i)T^{2} \)
7 \( 1 + (-1.11 + 0.742i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (-0.181 + 0.0550i)T + (9.14 - 6.11i)T^{2} \)
13 \( 1 + (-0.374 - 0.307i)T + (2.53 + 12.7i)T^{2} \)
17 \( 1 + (-1.79 + 0.742i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (-0.487 - 4.94i)T + (-18.6 + 3.70i)T^{2} \)
23 \( 1 + (5.40 - 1.07i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-2.45 + 8.10i)T + (-24.1 - 16.1i)T^{2} \)
31 \( 1 + (3.30 + 3.30i)T + 31iT^{2} \)
37 \( 1 + (-8.32 - 0.820i)T + (36.2 + 7.21i)T^{2} \)
41 \( 1 + (-0.580 - 2.91i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (-8.36 - 4.47i)T + (23.8 + 35.7i)T^{2} \)
47 \( 1 + (-3.57 - 8.62i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (3.12 + 10.2i)T + (-44.0 + 29.4i)T^{2} \)
59 \( 1 + (-10.8 + 8.90i)T + (11.5 - 57.8i)T^{2} \)
61 \( 1 + (-0.339 - 0.635i)T + (-33.8 + 50.7i)T^{2} \)
67 \( 1 + (-0.0570 - 0.106i)T + (-37.2 + 55.7i)T^{2} \)
71 \( 1 + (8.81 + 13.1i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (1.93 + 1.29i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-3.11 + 7.51i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (14.1 - 1.39i)T + (81.4 - 16.1i)T^{2} \)
89 \( 1 + (1.57 + 0.313i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (11.0 + 11.0i)T + 97iT^{2} \)
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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

−11.02259421193736523582142962784, −10.10039063375868802840903437425, −9.586750438461754677681901365019, −8.064113193432813495652771754279, −7.60461570613809042790232273962, −6.19802047269921728315841087055, −5.75660314387048805803261375377, −4.28755067737302941590630445026, −2.88538607045695326639084419311, −1.90107641187155588071170328224, 1.10092329653888826454355873237, 2.54026095931325341443869506667, 4.09026153001448460981509610658, 5.30171303313423008125713937103, 5.83744958653188020272204473317, 7.06095199624716430300530185558, 8.478839055005259434545500772497, 8.904399477292179249738673040309, 9.671849003064981139996048971099, 10.69018163832551124440752986533

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