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Properties

Label 2-2e9-128.109-c1-0-12
Degree $2$
Conductor $512$
Sign $0.931 + 0.362i$
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  + (1.98 + 1.06i)3-s + (2.06 − 2.51i)5-s + (−3.55 − 2.37i)7-s + (1.15 + 1.72i)9-s + (4.79 + 1.45i)11-s + (0.194 − 0.159i)13-s + (6.77 − 2.80i)15-s + (−2.23 − 0.924i)17-s + (0.0645 − 0.655i)19-s + (−4.53 − 8.48i)21-s + (5.38 + 1.07i)23-s + (−1.09 − 5.49i)25-s + (−0.204 − 2.07i)27-s + (1.65 + 5.46i)29-s + (−1.88 + 1.88i)31-s + ⋯
L(s)  = 1  + (1.14 + 0.613i)3-s + (0.923 − 1.12i)5-s + (−1.34 − 0.897i)7-s + (0.383 + 0.574i)9-s + (1.44 + 0.438i)11-s + (0.0539 − 0.0442i)13-s + (1.74 − 0.724i)15-s + (−0.541 − 0.224i)17-s + (0.0148 − 0.150i)19-s + (−0.989 − 1.85i)21-s + (1.12 + 0.223i)23-s + (−0.218 − 1.09i)25-s + (−0.0393 − 0.399i)27-s + (0.307 + 1.01i)29-s + (−0.338 + 0.338i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.362i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.931 + 0.362i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 0.931 + 0.362i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15127 - 0.403601i\)
\(L(\frac12)\) \(\approx\) \(2.15127 - 0.403601i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.98 - 1.06i)T + (1.66 + 2.49i)T^{2} \)
5 \( 1 + (-2.06 + 2.51i)T + (-0.975 - 4.90i)T^{2} \)
7 \( 1 + (3.55 + 2.37i)T + (2.67 + 6.46i)T^{2} \)
11 \( 1 + (-4.79 - 1.45i)T + (9.14 + 6.11i)T^{2} \)
13 \( 1 + (-0.194 + 0.159i)T + (2.53 - 12.7i)T^{2} \)
17 \( 1 + (2.23 + 0.924i)T + (12.0 + 12.0i)T^{2} \)
19 \( 1 + (-0.0645 + 0.655i)T + (-18.6 - 3.70i)T^{2} \)
23 \( 1 + (-5.38 - 1.07i)T + (21.2 + 8.80i)T^{2} \)
29 \( 1 + (-1.65 - 5.46i)T + (-24.1 + 16.1i)T^{2} \)
31 \( 1 + (1.88 - 1.88i)T - 31iT^{2} \)
37 \( 1 + (5.31 - 0.523i)T + (36.2 - 7.21i)T^{2} \)
41 \( 1 + (1.06 - 5.36i)T + (-37.8 - 15.6i)T^{2} \)
43 \( 1 + (3.35 - 1.79i)T + (23.8 - 35.7i)T^{2} \)
47 \( 1 + (2.40 - 5.80i)T + (-33.2 - 33.2i)T^{2} \)
53 \( 1 + (-2.52 + 8.31i)T + (-44.0 - 29.4i)T^{2} \)
59 \( 1 + (-4.78 - 3.92i)T + (11.5 + 57.8i)T^{2} \)
61 \( 1 + (2.32 - 4.34i)T + (-33.8 - 50.7i)T^{2} \)
67 \( 1 + (3.94 - 7.38i)T + (-37.2 - 55.7i)T^{2} \)
71 \( 1 + (1.89 - 2.83i)T + (-27.1 - 65.5i)T^{2} \)
73 \( 1 + (-2.61 + 1.75i)T + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (4.38 + 10.5i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (11.3 + 1.12i)T + (81.4 + 16.1i)T^{2} \)
89 \( 1 + (-4.78 + 0.952i)T + (82.2 - 34.0i)T^{2} \)
97 \( 1 + (-7.32 + 7.32i)T - 97iT^{2} \)
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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

−10.37839672394112099140736388102, −9.685196841473042609585016157697, −9.109402559185607350201441235032, −8.709829651194398496800894750741, −7.12383150295849446845904175198, −6.37121979054791740944061400542, −4.92430102272035400642331520725, −3.96171112330505341486121547208, −3.03732779334279785745125642160, −1.36309155294010798832598692480, 1.97693688267015316420475465121, 2.85685138007659354893190424051, 3.62696358966764436810520368905, 5.75939747776823551664623116889, 6.57884977102916078911308804137, 7.01096920843800244943007670642, 8.532838359392338809300371645389, 9.143882084429389732682863748843, 9.772802800918275928286266854455, 10.84138636638409634720217444750

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