Properties

Label 2-460-23.9-c1-0-7
Degree $2$
Conductor $460$
Sign $-0.966 + 0.257i$
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  + (−1.26 − 1.46i)3-s + (−0.142 − 0.989i)5-s + (0.0191 − 0.0418i)7-s + (−0.107 + 0.746i)9-s + (0.229 + 0.0674i)11-s + (−1.75 − 3.83i)13-s + (−1.26 + 1.46i)15-s + (−0.597 − 0.383i)17-s + (−3.34 + 2.14i)19-s + (−0.0856 + 0.0251i)21-s + (−4.35 − 2.00i)23-s + (−0.959 + 0.281i)25-s + (−3.66 + 2.35i)27-s + (−0.590 − 0.379i)29-s + (−3.43 + 3.96i)31-s + ⋯
L(s)  = 1  + (−0.732 − 0.845i)3-s + (−0.0636 − 0.442i)5-s + (0.00723 − 0.0158i)7-s + (−0.0357 + 0.248i)9-s + (0.0692 + 0.0203i)11-s + (−0.485 − 1.06i)13-s + (−0.327 + 0.378i)15-s + (−0.144 − 0.0930i)17-s + (−0.766 + 0.492i)19-s + (−0.0186 + 0.00548i)21-s + (−0.908 − 0.417i)23-s + (−0.191 + 0.0563i)25-s + (−0.704 + 0.452i)27-s + (−0.109 − 0.0704i)29-s + (−0.617 + 0.712i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.966 + 0.257i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ -0.966 + 0.257i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0825898 - 0.631637i\)
\(L(\frac12)\) \(\approx\) \(0.0825898 - 0.631637i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.142 + 0.989i)T \)
23 \( 1 + (4.35 + 2.00i)T \)
good3 \( 1 + (1.26 + 1.46i)T + (-0.426 + 2.96i)T^{2} \)
7 \( 1 + (-0.0191 + 0.0418i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (-0.229 - 0.0674i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (1.75 + 3.83i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (0.597 + 0.383i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (3.34 - 2.14i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (0.590 + 0.379i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (3.43 - 3.96i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-0.347 + 2.41i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (0.510 + 3.55i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (3.44 + 3.97i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + 0.0114T + 47T^{2} \)
53 \( 1 + (-5.16 + 11.3i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (2.03 + 4.45i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (-3.47 + 4.01i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (-2.04 + 0.599i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (-7.02 + 2.06i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (0.786 - 0.505i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-4.12 - 9.03i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (-1.46 + 10.2i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (-1.17 - 1.35i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (0.0699 + 0.486i)T + (-93.0 + 27.3i)T^{2} \)
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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

−10.74140188695569822175599675479, −9.877638039813394808075596846021, −8.690681978633750363328901846673, −7.81065015082353765915687213438, −6.89040471367440605842792709241, −5.94689794384113072062857014576, −5.12040050101071654329693708306, −3.73423905169882131674906695515, −1.99982120561146105843800862598, −0.41366832324406608780235736168, 2.22822279154931786444936253954, 3.89863100400416437440774351156, 4.67026591706311312434893453074, 5.78592589036053907526075944606, 6.70714294575405661030302858859, 7.75736653741611046926933915025, 9.008882638800262496229920822882, 9.853660816903410129541684681479, 10.59197201129091028894648462664, 11.40575176665329613741100054402

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