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Properties

Label 2-460-23.3-c1-0-0
Degree $2$
Conductor $460$
Sign $-0.528 + 0.849i$
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  + (−1.33 + 2.92i)3-s + (0.654 − 0.755i)5-s + (−3.81 + 2.44i)7-s + (−4.81 − 5.55i)9-s + (0.397 − 2.76i)11-s + (0.0567 + 0.0364i)13-s + (1.33 + 2.92i)15-s + (0.473 + 0.138i)17-s + (−7.47 + 2.19i)19-s + (−2.07 − 14.4i)21-s + (4.70 − 0.925i)23-s + (−0.142 − 0.989i)25-s + (13.4 − 3.94i)27-s + (1.47 + 0.434i)29-s + (−0.544 − 1.19i)31-s + ⋯
L(s)  = 1  + (−0.771 + 1.68i)3-s + (0.292 − 0.337i)5-s + (−1.44 + 0.925i)7-s + (−1.60 − 1.85i)9-s + (0.119 − 0.834i)11-s + (0.0157 + 0.0101i)13-s + (0.345 + 0.755i)15-s + (0.114 + 0.0336i)17-s + (−1.71 + 0.503i)19-s + (−0.452 − 3.14i)21-s + (0.981 − 0.192i)23-s + (−0.0284 − 0.197i)25-s + (2.58 − 0.759i)27-s + (0.274 + 0.0806i)29-s + (−0.0978 − 0.214i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.528 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.528 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.528 + 0.849i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ -0.528 + 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0791279 - 0.142440i\)
\(L(\frac12)\) \(\approx\) \(0.0791279 - 0.142440i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.654 + 0.755i)T \)
23 \( 1 + (-4.70 + 0.925i)T \)
good3 \( 1 + (1.33 - 2.92i)T + (-1.96 - 2.26i)T^{2} \)
7 \( 1 + (3.81 - 2.44i)T + (2.90 - 6.36i)T^{2} \)
11 \( 1 + (-0.397 + 2.76i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (-0.0567 - 0.0364i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (-0.473 - 0.138i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (7.47 - 2.19i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (-1.47 - 0.434i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (0.544 + 1.19i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (5.83 + 6.73i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (2.22 - 2.57i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (-0.0515 + 0.112i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + (10.8 - 6.94i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (3.09 + 1.99i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (-5.02 - 10.9i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (-1.75 - 12.1i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (1.54 + 10.7i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (7.32 - 2.15i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (4.86 + 3.12i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (-1.97 - 2.28i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (3.77 - 8.27i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (2.76 - 3.19i)T + (-13.8 - 96.0i)T^{2} \)
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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

−11.45059332479172406235225579615, −10.58346542496837102706838003720, −9.939903771090704190414669158786, −9.046006076167012255459395889072, −8.669721687117885223775470259446, −6.43821027663552948758315843593, −5.96852714458865765776582554562, −5.05795881135921383562982842028, −3.89250386285342135019849677738, −2.94490037754735187811196081049, 0.10509769078718906625516006083, 1.75112094323278461142803116590, 3.12283382518837769765854889267, 4.85455462818462658951303293180, 6.28425745842483854471218140862, 6.71124973479874875591483276092, 7.23375449222406311115218138725, 8.425228064191087737347754969821, 9.761001685949030673125193776043, 10.60575163530213977622574526389

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