Properties

Label 2-460-23.13-c1-0-0
Degree $2$
Conductor $460$
Sign $-0.555 - 0.831i$
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  + (−2.04 − 1.31i)3-s + (0.415 − 0.909i)5-s + (−2.79 − 0.819i)7-s + (1.20 + 2.62i)9-s + (−0.00424 + 0.00490i)11-s + (−1.23 + 0.362i)13-s + (−2.04 + 1.31i)15-s + (0.254 + 1.76i)17-s + (−0.387 + 2.69i)19-s + (4.62 + 5.33i)21-s + (−4.79 + 0.0277i)23-s + (−0.654 − 0.755i)25-s + (−0.0378 + 0.263i)27-s + (1.00 + 7.01i)29-s + (−1.95 + 1.25i)31-s + ⋯
L(s)  = 1  + (−1.17 − 0.757i)3-s + (0.185 − 0.406i)5-s + (−1.05 − 0.309i)7-s + (0.400 + 0.876i)9-s + (−0.00128 + 0.00147i)11-s + (−0.342 + 0.100i)13-s + (−0.527 + 0.338i)15-s + (0.0616 + 0.428i)17-s + (−0.0890 + 0.619i)19-s + (1.00 + 1.16i)21-s + (−0.999 + 0.00577i)23-s + (−0.130 − 0.151i)25-s + (−0.00728 + 0.0506i)27-s + (0.187 + 1.30i)29-s + (−0.351 + 0.226i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.555 - 0.831i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ -0.555 - 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0162976 + 0.0304799i\)
\(L(\frac12)\) \(\approx\) \(0.0162976 + 0.0304799i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (4.79 - 0.0277i)T \)
good3 \( 1 + (2.04 + 1.31i)T + (1.24 + 2.72i)T^{2} \)
7 \( 1 + (2.79 + 0.819i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (0.00424 - 0.00490i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (1.23 - 0.362i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (-0.254 - 1.76i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (0.387 - 2.69i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-1.00 - 7.01i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (1.95 - 1.25i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-2.12 - 4.64i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (-0.286 + 0.627i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (8.33 + 5.35i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 + 0.275T + 47T^{2} \)
53 \( 1 + (8.57 + 2.51i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (-0.409 + 0.120i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (4.69 - 3.01i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (-7.40 - 8.55i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (5.36 + 6.18i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-1.57 + 10.9i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (-5.51 + 1.61i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (2.44 + 5.34i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (10.3 + 6.67i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (-5.22 + 11.4i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.53815134920159970041412996877, −10.46084731905940344187851061027, −9.813664097434778168465638349461, −8.629781826525153055798244218413, −7.44662918692894133302412532221, −6.55771853703336214618878970589, −5.94160818780846068272486254834, −4.90447419954823167888549212127, −3.46902299038208375686261870229, −1.60031631784332527149399170859, 0.02414498226284977216367591006, 2.63937140664237319886806670735, 3.98331410644157555097723351520, 5.10428082741422770868002402126, 6.04730806297262511831470627587, 6.65006963097679985410313470715, 7.954819760569103705771952987630, 9.527184630144277057330901969668, 9.775863857146882624391486997145, 10.77697772769381700683552477562

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