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Properties

Label 2-2e9-16.13-c1-0-14
Degree $2$
Conductor $512$
Sign $-0.707 + 0.707i$
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  + (−0.765 − 0.765i)3-s + (0.414 − 0.414i)5-s − 3.69i·7-s − 1.82i·9-s + (−2.93 + 2.93i)11-s + (2.41 + 2.41i)13-s − 0.634·15-s − 2.82·17-s + (−4.46 − 4.46i)19-s + (−2.82 + 2.82i)21-s − 6.75i·23-s + 4.65i·25-s + (−3.69 + 3.69i)27-s + (−5.24 − 5.24i)29-s + 3.06·31-s + ⋯
L(s)  = 1  + (−0.441 − 0.441i)3-s + (0.185 − 0.185i)5-s − 1.39i·7-s − 0.609i·9-s + (−0.883 + 0.883i)11-s + (0.669 + 0.669i)13-s − 0.163·15-s − 0.685·17-s + (−1.02 − 1.02i)19-s + (−0.617 + 0.617i)21-s − 1.40i·23-s + 0.931i·25-s + (−0.711 + 0.711i)27-s + (−0.973 − 0.973i)29-s + 0.549·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.338755 - 0.817827i\)
\(L(\frac12)\) \(\approx\) \(0.338755 - 0.817827i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.765 + 0.765i)T + 3iT^{2} \)
5 \( 1 + (-0.414 + 0.414i)T - 5iT^{2} \)
7 \( 1 + 3.69iT - 7T^{2} \)
11 \( 1 + (2.93 - 2.93i)T - 11iT^{2} \)
13 \( 1 + (-2.41 - 2.41i)T + 13iT^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 + (4.46 + 4.46i)T + 19iT^{2} \)
23 \( 1 + 6.75iT - 23T^{2} \)
29 \( 1 + (5.24 + 5.24i)T + 29iT^{2} \)
31 \( 1 - 3.06T + 31T^{2} \)
37 \( 1 + (-6.41 + 6.41i)T - 37iT^{2} \)
41 \( 1 + 4iT - 41T^{2} \)
43 \( 1 + (-0.765 + 0.765i)T - 43iT^{2} \)
47 \( 1 + 3.06T + 47T^{2} \)
53 \( 1 + (3.24 - 3.24i)T - 53iT^{2} \)
59 \( 1 + (-0.765 + 0.765i)T - 59iT^{2} \)
61 \( 1 + (-0.757 - 0.757i)T + 61iT^{2} \)
67 \( 1 + (1.39 + 1.39i)T + 67iT^{2} \)
71 \( 1 - 8.02iT - 71T^{2} \)
73 \( 1 + 6.48iT - 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 + (-9.68 - 9.68i)T + 83iT^{2} \)
89 \( 1 + 4.82iT - 89T^{2} \)
97 \( 1 - 5.17T + 97T^{2} \)
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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

−10.79596580431882191765928631859, −9.718291765157065262010554172556, −8.851022512382501846223684025922, −7.60767107384983216059848173959, −6.87962380681795761576396305527, −6.16346439540027830422382183128, −4.70408731845672587016690169169, −3.95866571589529200275521969013, −2.15005906629136943736267029243, −0.53312798155045654810628430163, 2.13394192043450408939124189694, 3.30409461289165851670123381540, 4.83146288552174148118938395691, 5.72998808183196530456845916945, 6.19883658891310399142598048243, 7.965986361438618368600296293170, 8.423055377841852085854271382393, 9.542720123122030053893184427349, 10.52784068339956822925601783277, 11.07846449815067100653851397733

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