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Properties

Label 2-464-116.115-c2-0-12
Degree $2$
Conductor $464$
Sign $0.435 + 0.900i$
Analytic cond. $12.6430$
Root an. cond. $3.55571$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.54·3-s − 5.75·5-s + 7.46i·7-s + 3.55·9-s + 9.81·11-s − 24.4·13-s + 20.4·15-s + 31.6i·17-s − 10.6·19-s − 26.4i·21-s − 31.8i·23-s + 8.16·25-s + 19.2·27-s + (16.2 − 23.9i)29-s + 43.2·31-s + ⋯
L(s)  = 1  − 1.18·3-s − 1.15·5-s + 1.06i·7-s + 0.394·9-s + 0.892·11-s − 1.88·13-s + 1.36·15-s + 1.86i·17-s − 0.562·19-s − 1.25i·21-s − 1.38i·23-s + 0.326·25-s + 0.714·27-s + (0.561 − 0.827i)29-s + 1.39·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(464\)    =    \(2^{4} \cdot 29\)
Sign: $0.435 + 0.900i$
Analytic conductor: \(12.6430\)
Root analytic conductor: \(3.55571\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{464} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 464,\ (\ :1),\ 0.435 + 0.900i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3806265127\)
\(L(\frac12)\) \(\approx\) \(0.3806265127\)
\(L(2)\) 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 \( 1 + (-16.2 + 23.9i)T \)
good3 \( 1 + 3.54T + 9T^{2} \)
5 \( 1 + 5.75T + 25T^{2} \)
7 \( 1 - 7.46iT - 49T^{2} \)
11 \( 1 - 9.81T + 121T^{2} \)
13 \( 1 + 24.4T + 169T^{2} \)
17 \( 1 - 31.6iT - 289T^{2} \)
19 \( 1 + 10.6T + 361T^{2} \)
23 \( 1 + 31.8iT - 529T^{2} \)
31 \( 1 - 43.2T + 961T^{2} \)
37 \( 1 + 31.4iT - 1.36e3T^{2} \)
41 \( 1 + 58.5iT - 1.68e3T^{2} \)
43 \( 1 - 0.780T + 1.84e3T^{2} \)
47 \( 1 + 58.1T + 2.20e3T^{2} \)
53 \( 1 + 6.29T + 2.80e3T^{2} \)
59 \( 1 - 79.3iT - 3.48e3T^{2} \)
61 \( 1 + 97.5iT - 3.72e3T^{2} \)
67 \( 1 + 61.4iT - 4.48e3T^{2} \)
71 \( 1 - 72.5iT - 5.04e3T^{2} \)
73 \( 1 - 65.6iT - 5.32e3T^{2} \)
79 \( 1 - 152.T + 6.24e3T^{2} \)
83 \( 1 + 13.4iT - 6.88e3T^{2} \)
89 \( 1 - 44.0iT - 7.92e3T^{2} \)
97 \( 1 + 26.8iT - 9.40e3T^{2} \)
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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

−10.85021329111716239842722280951, −9.992135859909670308924676476475, −8.722257407704124537193636467712, −8.010449685762454877295557830781, −6.71975528481630778607784228085, −6.05235907248709838862836570845, −4.89280187771612941276616378732, −4.05959608390791766789301137617, −2.37008638733329363992891206703, −0.26664419552179774332153009212, 0.828489051085108487486403810277, 3.13624471496315553461419209158, 4.53377319890528997869277400217, 4.96834720508582417032144504366, 6.57709244385942404552143443107, 7.17854560189193665204960194186, 7.964957712183780052117070154734, 9.458504133826006555674324002639, 10.19941966739331172497575355443, 11.31601055345491366661248824426

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