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Properties

Label 2-2e9-128.101-c1-0-0
Degree $2$
Conductor $512$
Sign $-0.765 - 0.643i$
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  + (−0.205 + 0.109i)3-s + (−1.29 − 1.58i)5-s + (−1.60 + 1.06i)7-s + (−1.63 + 2.44i)9-s + (2.28 − 0.693i)11-s + (−1.01 − 0.828i)13-s + (0.440 + 0.182i)15-s + (−6.19 + 2.56i)17-s + (0.717 + 7.28i)19-s + (0.211 − 0.395i)21-s + (−6.82 + 1.35i)23-s + (0.159 − 0.804i)25-s + (0.135 − 1.37i)27-s + (−1.11 + 3.69i)29-s + (0.0726 + 0.0726i)31-s + ⋯
L(s)  = 1  + (−0.118 + 0.0634i)3-s + (−0.580 − 0.706i)5-s + (−0.604 + 0.404i)7-s + (−0.545 + 0.816i)9-s + (0.688 − 0.208i)11-s + (−0.280 − 0.229i)13-s + (0.113 + 0.0471i)15-s + (−1.50 + 0.621i)17-s + (0.164 + 1.67i)19-s + (0.0461 − 0.0863i)21-s + (−1.42 + 0.282i)23-s + (0.0319 − 0.160i)25-s + (0.0261 − 0.265i)27-s + (−0.207 + 0.685i)29-s + (0.0130 + 0.0130i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 - 0.643i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $-0.765 - 0.643i$
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: \((2,\ 512,\ (\ :1/2),\ -0.765 - 0.643i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.132444 + 0.363638i\)
\(L(\frac12)\) \(\approx\) \(0.132444 + 0.363638i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.205 - 0.109i)T + (1.66 - 2.49i)T^{2} \)
5 \( 1 + (1.29 + 1.58i)T + (-0.975 + 4.90i)T^{2} \)
7 \( 1 + (1.60 - 1.06i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (-2.28 + 0.693i)T + (9.14 - 6.11i)T^{2} \)
13 \( 1 + (1.01 + 0.828i)T + (2.53 + 12.7i)T^{2} \)
17 \( 1 + (6.19 - 2.56i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (-0.717 - 7.28i)T + (-18.6 + 3.70i)T^{2} \)
23 \( 1 + (6.82 - 1.35i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (1.11 - 3.69i)T + (-24.1 - 16.1i)T^{2} \)
31 \( 1 + (-0.0726 - 0.0726i)T + 31iT^{2} \)
37 \( 1 + (-2.87 - 0.282i)T + (36.2 + 7.21i)T^{2} \)
41 \( 1 + (-0.658 - 3.30i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (3.06 + 1.63i)T + (23.8 + 35.7i)T^{2} \)
47 \( 1 + (1.75 + 4.23i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (2.91 + 9.59i)T + (-44.0 + 29.4i)T^{2} \)
59 \( 1 + (4.93 - 4.04i)T + (11.5 - 57.8i)T^{2} \)
61 \( 1 + (0.733 + 1.37i)T + (-33.8 + 50.7i)T^{2} \)
67 \( 1 + (2.54 + 4.75i)T + (-37.2 + 55.7i)T^{2} \)
71 \( 1 + (1.60 + 2.39i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-11.8 - 7.92i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (3.13 - 7.56i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-10.8 + 1.06i)T + (81.4 - 16.1i)T^{2} \)
89 \( 1 + (-12.0 - 2.38i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (2.03 + 2.03i)T + 97iT^{2} \)
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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

−11.33800905000769582871796306158, −10.36542155276541601329042483270, −9.417963467756951123344993466434, −8.418769534832008349393093981938, −7.966258040236638330912048931924, −6.50306996803602312269332994756, −5.70512609206626077395782265040, −4.54720501564157527416382898106, −3.56083814374621746612907495986, −1.97415340837877228261246848457, 0.21944606174914145246371095050, 2.55368318982984299885783932522, 3.67050300997332689102608199139, 4.63205629204703846914705225090, 6.31107640613943074378693480356, 6.74234118991406906338885748820, 7.64035835911573045748320029218, 9.027023745724379500614513602122, 9.487967179596626182499370948948, 10.73098515109383020676521811136

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