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Properties

Label 2-464-116.15-c1-0-1
Degree $2$
Conductor $464$
Sign $0.00948 - 0.999i$
Analytic cond. $3.70505$
Root an. cond. $1.92485$
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  + (−1.77 − 2.82i)3-s + (0.925 + 0.738i)5-s + (−2.85 − 0.652i)7-s + (−3.52 + 7.31i)9-s + (−1.40 + 4.02i)11-s + (−0.443 − 0.920i)13-s + (0.442 − 3.92i)15-s + (−2.46 − 2.46i)17-s + (4.11 + 2.58i)19-s + (3.23 + 9.23i)21-s + (−4.63 + 3.69i)23-s + (−0.800 − 3.50i)25-s + (16.9 − 1.91i)27-s + (5.09 + 1.73i)29-s + (0.225 + 2.00i)31-s + ⋯
L(s)  = 1  + (−1.02 − 1.63i)3-s + (0.413 + 0.330i)5-s + (−1.08 − 0.246i)7-s + (−1.17 + 2.43i)9-s + (−0.424 + 1.21i)11-s + (−0.122 − 0.255i)13-s + (0.114 − 1.01i)15-s + (−0.597 − 0.597i)17-s + (0.944 + 0.593i)19-s + (0.704 + 2.01i)21-s + (−0.966 + 0.770i)23-s + (−0.160 − 0.701i)25-s + (3.26 − 0.367i)27-s + (0.946 + 0.322i)29-s + (0.0405 + 0.359i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00948 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00948 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(464\)    =    \(2^{4} \cdot 29\)
Sign: $0.00948 - 0.999i$
Analytic conductor: \(3.70505\)
Root analytic conductor: \(1.92485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{464} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 464,\ (\ :1/2),\ 0.00948 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.160402 + 0.158889i\)
\(L(\frac12)\) \(\approx\) \(0.160402 + 0.158889i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (-5.09 - 1.73i)T \)
good3 \( 1 + (1.77 + 2.82i)T + (-1.30 + 2.70i)T^{2} \)
5 \( 1 + (-0.925 - 0.738i)T + (1.11 + 4.87i)T^{2} \)
7 \( 1 + (2.85 + 0.652i)T + (6.30 + 3.03i)T^{2} \)
11 \( 1 + (1.40 - 4.02i)T + (-8.60 - 6.85i)T^{2} \)
13 \( 1 + (0.443 + 0.920i)T + (-8.10 + 10.1i)T^{2} \)
17 \( 1 + (2.46 + 2.46i)T + 17iT^{2} \)
19 \( 1 + (-4.11 - 2.58i)T + (8.24 + 17.1i)T^{2} \)
23 \( 1 + (4.63 - 3.69i)T + (5.11 - 22.4i)T^{2} \)
31 \( 1 + (-0.225 - 2.00i)T + (-30.2 + 6.89i)T^{2} \)
37 \( 1 + (10.8 - 3.78i)T + (28.9 - 23.0i)T^{2} \)
41 \( 1 + (4.85 - 4.85i)T - 41iT^{2} \)
43 \( 1 + (-4.38 - 0.493i)T + (41.9 + 9.56i)T^{2} \)
47 \( 1 + (2.59 + 0.909i)T + (36.7 + 29.3i)T^{2} \)
53 \( 1 + (4.79 - 6.01i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + 4.30iT - 59T^{2} \)
61 \( 1 + (-3.63 + 2.28i)T + (26.4 - 54.9i)T^{2} \)
67 \( 1 + (3.97 + 1.91i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (1.63 - 0.789i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-2.98 - 0.336i)T + (71.1 + 16.2i)T^{2} \)
79 \( 1 + (14.0 - 4.90i)T + (61.7 - 49.2i)T^{2} \)
83 \( 1 + (8.15 - 1.86i)T + (74.7 - 36.0i)T^{2} \)
89 \( 1 + (-6.41 + 0.722i)T + (86.7 - 19.8i)T^{2} \)
97 \( 1 + (14.8 + 9.32i)T + (42.0 + 87.3i)T^{2} \)
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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

−11.52608897603778859553812905209, −10.32806835848801746519251270281, −9.830292697964222507185518533832, −8.200739827288785709256911648931, −7.22971313021887820539222908655, −6.75100959489134594117708154344, −5.90317730332710378803588461888, −4.92047847768419609690097453783, −2.88488461190306930255125734212, −1.66440153165561715521856865331, 0.15159288531644226800590404508, 3.06218943604457533425007952151, 4.02886671743848293865800646509, 5.21637209434496936374605903719, 5.86305635908859472680843473838, 6.64732728554043345303154366896, 8.599195265210919257461555077637, 9.235165406604146269743621666522, 10.04450006110795694255321759352, 10.64757986501833265215258328937

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