Properties

Label 2-2e9-32.13-c1-0-6
Degree $2$
Conductor $512$
Sign $0.980 + 0.195i$
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  + (0.292 − 0.707i)3-s + (2.70 − 1.12i)5-s + (−1 + i)7-s + (1.70 + 1.70i)9-s + (1.70 + 4.12i)11-s + (0.707 + 0.292i)13-s − 2.24i·15-s − 2.82i·17-s + (3.70 + 1.53i)19-s + (0.414 + i)21-s + (−5.82 − 5.82i)23-s + (2.53 − 2.53i)25-s + (3.82 − 1.58i)27-s + (1.29 − 3.12i)29-s − 4·31-s + ⋯
L(s)  = 1  + (0.169 − 0.408i)3-s + (1.21 − 0.501i)5-s + (−0.377 + 0.377i)7-s + (0.569 + 0.569i)9-s + (0.514 + 1.24i)11-s + (0.196 + 0.0812i)13-s − 0.579i·15-s − 0.685i·17-s + (0.850 + 0.352i)19-s + (0.0903 + 0.218i)21-s + (−1.21 − 1.21i)23-s + (0.507 − 0.507i)25-s + (0.736 − 0.305i)27-s + (0.240 − 0.579i)29-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.980 + 0.195i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 0.980 + 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88255 - 0.185415i\)
\(L(\frac12)\) \(\approx\) \(1.88255 - 0.185415i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.292 + 0.707i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (-2.70 + 1.12i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 + (-1.70 - 4.12i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.707 - 0.292i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 + (-3.70 - 1.53i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (5.82 + 5.82i)T + 23iT^{2} \)
29 \( 1 + (-1.29 + 3.12i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-0.707 + 0.292i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.171 - 0.171i)T + 41iT^{2} \)
43 \( 1 + (1.94 + 4.70i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 0.343iT - 47T^{2} \)
53 \( 1 + (-0.464 - 1.12i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (4.53 - 1.87i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (0.707 - 1.70i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-2.29 + 5.53i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-5.82 + 5.82i)T - 71iT^{2} \)
73 \( 1 + (7 + 7i)T + 73iT^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 + (-4.53 - 1.87i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (8.65 - 8.65i)T - 89iT^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−10.65048584710416217919576027492, −9.705216442706347920129763091629, −9.409273535525351113154095939776, −8.180192406596197394810109740948, −7.16783059579153792762370017628, −6.28314833168372276568282375420, −5.28722972975958068411644569702, −4.27505666431580358242657037588, −2.45425989145403632854503248510, −1.58404581371373945712015655848, 1.44977343637243623639026218054, 3.14384147017070970659978491641, 3.91543387539799411328306867754, 5.53664352247656033853122813304, 6.24037218533497487655604270079, 7.09312695549494287295714707077, 8.423188524801275616672389895357, 9.464308094401005583175392853577, 9.880204974515663810472311247959, 10.74083597695335746695095347736

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