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Properties

Label 2-460-5.4-c1-0-4
Degree $2$
Conductor $460$
Sign $0.732 - 0.680i$
Analytic cond. $3.67311$
Root an. cond. $1.91653$
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  − 0.486i·3-s + (1.52 + 1.63i)5-s + 1.80i·7-s + 2.76·9-s − 2.90·11-s + 2.25i·13-s + (0.796 − 0.740i)15-s + 2.14i·17-s − 0.339·19-s + 0.877·21-s + i·23-s + (−0.369 + 4.98i)25-s − 2.80i·27-s + 5.60·29-s + 5.92·31-s + ⋯
L(s)  = 1  − 0.280i·3-s + (0.680 + 0.732i)5-s + 0.682i·7-s + 0.921·9-s − 0.876·11-s + 0.624i·13-s + (0.205 − 0.191i)15-s + 0.520i·17-s − 0.0779·19-s + 0.191·21-s + 0.208i·23-s + (−0.0738 + 0.997i)25-s − 0.539i·27-s + 1.04·29-s + 1.06·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.732 - 0.680i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.732 - 0.680i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46691 + 0.576076i\)
\(L(\frac12)\) \(\approx\) \(1.46691 + 0.576076i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.52 - 1.63i)T \)
23 \( 1 - iT \)
good3 \( 1 + 0.486iT - 3T^{2} \)
7 \( 1 - 1.80iT - 7T^{2} \)
11 \( 1 + 2.90T + 11T^{2} \)
13 \( 1 - 2.25iT - 13T^{2} \)
17 \( 1 - 2.14iT - 17T^{2} \)
19 \( 1 + 0.339T + 19T^{2} \)
29 \( 1 - 5.60T + 29T^{2} \)
31 \( 1 - 5.92T + 31T^{2} \)
37 \( 1 + 8.98iT - 37T^{2} \)
41 \( 1 - 1.89T + 41T^{2} \)
43 \( 1 - 9.47iT - 43T^{2} \)
47 \( 1 + 7.83iT - 47T^{2} \)
53 \( 1 + 6.47iT - 53T^{2} \)
59 \( 1 + 5.17T + 59T^{2} \)
61 \( 1 + 9.12T + 61T^{2} \)
67 \( 1 + 9.25iT - 67T^{2} \)
71 \( 1 + 4.60T + 71T^{2} \)
73 \( 1 - 11.3iT - 73T^{2} \)
79 \( 1 + 7.94T + 79T^{2} \)
83 \( 1 + 5.37iT - 83T^{2} \)
89 \( 1 + 12.9T + 89T^{2} \)
97 \( 1 + 2.43iT - 97T^{2} \)
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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

−11.04751638218307184209238321067, −10.22968851512237929779192252933, −9.543560603761327150794610656915, −8.427334285085218336600353697592, −7.38482366494335465654778680034, −6.51684151512512786454070664056, −5.66512027349957840520776165589, −4.42257437801403318740455368773, −2.87114910420093116605397656773, −1.82304006572748199700763531543, 1.09393980449437093455410529089, 2.77823605989544294491046751379, 4.36668510677589635708453071033, 5.02685027168987727640238583802, 6.20774690638937211290686713171, 7.34965062376967189909795801122, 8.241507418652367429755474080267, 9.286831667076524085891930872771, 10.32751473143650127867399456819, 10.41611134614326008166865671719

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