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Properties

Label 2-464-29.12-c2-0-19
Degree $2$
Conductor $464$
Sign $-0.298 + 0.954i$
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  + (0.417 + 0.417i)3-s + 2.93i·5-s − 2.28·7-s − 8.65i·9-s + (−10.9 − 10.9i)11-s + 6.59i·13-s + (−1.22 + 1.22i)15-s + (1.72 + 1.72i)17-s + (−23.4 − 23.4i)19-s + (−0.951 − 0.951i)21-s − 0.318·23-s + 16.3·25-s + (7.36 − 7.36i)27-s + (13.7 − 25.5i)29-s + (−11.3 − 11.3i)31-s + ⋯
L(s)  = 1  + (0.139 + 0.139i)3-s + 0.586i·5-s − 0.325·7-s − 0.961i·9-s + (−0.991 − 0.991i)11-s + 0.507i·13-s + (−0.0816 + 0.0816i)15-s + (0.101 + 0.101i)17-s + (−1.23 − 1.23i)19-s + (−0.0453 − 0.0453i)21-s − 0.0138·23-s + 0.655·25-s + (0.272 − 0.272i)27-s + (0.473 − 0.880i)29-s + (−0.367 − 0.367i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(464\)    =    \(2^{4} \cdot 29\)
Sign: $-0.298 + 0.954i$
Analytic conductor: \(12.6430\)
Root analytic conductor: \(3.55571\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{464} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 464,\ (\ :1),\ -0.298 + 0.954i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9161542639\)
\(L(\frac12)\) \(\approx\) \(0.9161542639\)
\(L(2)\) 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 \( 1 + (-13.7 + 25.5i)T \)
good3 \( 1 + (-0.417 - 0.417i)T + 9iT^{2} \)
5 \( 1 - 2.93iT - 25T^{2} \)
7 \( 1 + 2.28T + 49T^{2} \)
11 \( 1 + (10.9 + 10.9i)T + 121iT^{2} \)
13 \( 1 - 6.59iT - 169T^{2} \)
17 \( 1 + (-1.72 - 1.72i)T + 289iT^{2} \)
19 \( 1 + (23.4 + 23.4i)T + 361iT^{2} \)
23 \( 1 + 0.318T + 529T^{2} \)
31 \( 1 + (11.3 + 11.3i)T + 961iT^{2} \)
37 \( 1 + (-32.1 + 32.1i)T - 1.36e3iT^{2} \)
41 \( 1 + (21.4 - 21.4i)T - 1.68e3iT^{2} \)
43 \( 1 + (2.22 + 2.22i)T + 1.84e3iT^{2} \)
47 \( 1 + (-38.1 + 38.1i)T - 2.20e3iT^{2} \)
53 \( 1 + 85.3T + 2.80e3T^{2} \)
59 \( 1 + 80.6T + 3.48e3T^{2} \)
61 \( 1 + (46.7 + 46.7i)T + 3.72e3iT^{2} \)
67 \( 1 - 65.7iT - 4.48e3T^{2} \)
71 \( 1 + 31.4iT - 5.04e3T^{2} \)
73 \( 1 + (-52.9 + 52.9i)T - 5.32e3iT^{2} \)
79 \( 1 + (-55.5 - 55.5i)T + 6.24e3iT^{2} \)
83 \( 1 - 130.T + 6.88e3T^{2} \)
89 \( 1 + (-13.1 - 13.1i)T + 7.92e3iT^{2} \)
97 \( 1 + (-31.7 + 31.7i)T - 9.40e3iT^{2} \)
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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

−10.72707006019160101160196228155, −9.615205052223714413831376505148, −8.853888194277684636812734109296, −7.88032740458894886195421672615, −6.65717092064950042174562130524, −6.11364579504054577485704165900, −4.68219466864673265840652302657, −3.45080991750347973450554206678, −2.52363521197060814244876614520, −0.35484008626338777624330763799, 1.69364720613600184715001181724, 2.94670738391153734144248019062, 4.53894645542353350790870929286, 5.23834165168688491073152268978, 6.47052825652525544830791170460, 7.70532000265344768678315989801, 8.193932702794569746071117612162, 9.306085875319600877856007716721, 10.38197993214154071466051983135, 10.76948933373876070872695890408

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