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Properties

Label 2-460-23.4-c1-0-4
Degree $2$
Conductor $460$
Sign $0.998 + 0.0573i$
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  + (2.60 − 0.763i)3-s + (−0.841 − 0.540i)5-s + (0.742 + 5.16i)7-s + (3.65 − 2.35i)9-s + (2.19 − 4.81i)11-s + (−0.380 + 2.64i)13-s + (−2.60 − 0.763i)15-s + (2.62 + 3.02i)17-s + (3.33 − 3.84i)19-s + (5.87 + 12.8i)21-s + (−4.55 − 1.49i)23-s + (0.415 + 0.909i)25-s + (2.39 − 2.76i)27-s + (−5.18 − 5.98i)29-s + (−0.294 − 0.0864i)31-s + ⋯
L(s)  = 1  + (1.50 − 0.440i)3-s + (−0.376 − 0.241i)5-s + (0.280 + 1.95i)7-s + (1.21 − 0.783i)9-s + (0.663 − 1.45i)11-s + (−0.105 + 0.733i)13-s + (−0.671 − 0.197i)15-s + (0.635 + 0.733i)17-s + (0.764 − 0.881i)19-s + (1.28 + 2.80i)21-s + (−0.949 − 0.312i)23-s + (0.0830 + 0.181i)25-s + (0.460 − 0.531i)27-s + (−0.962 − 1.11i)29-s + (−0.0528 − 0.0155i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.998 + 0.0573i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.998 + 0.0573i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23638 - 0.0642346i\)
\(L(\frac12)\) \(\approx\) \(2.23638 - 0.0642346i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.841 + 0.540i)T \)
23 \( 1 + (4.55 + 1.49i)T \)
good3 \( 1 + (-2.60 + 0.763i)T + (2.52 - 1.62i)T^{2} \)
7 \( 1 + (-0.742 - 5.16i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (-2.19 + 4.81i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (0.380 - 2.64i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-2.62 - 3.02i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-3.33 + 3.84i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (5.18 + 5.98i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (0.294 + 0.0864i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (0.239 - 0.154i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (2.42 + 1.56i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (1.15 - 0.339i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + 8.10T + 47T^{2} \)
53 \( 1 + (0.852 + 5.92i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (1.14 - 7.93i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (-8.37 - 2.45i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (-1.58 - 3.47i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (-0.519 - 1.13i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (2.65 - 3.05i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (-0.818 + 5.69i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (1.06 - 0.681i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-14.0 + 4.13i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (8.72 + 5.60i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.49573918000825746527613319649, −9.669283946675154448094561174479, −8.939633750216665361389464555914, −8.483061333063755611886843741658, −7.81001019754311632586765328475, −6.40873693879739445484034927196, −5.44894775785988541177357538226, −3.84206624657165227094687622779, −2.86107023438925232468242933942, −1.79989970064410434841334938951, 1.61063097391017270900497216373, 3.36948176345755120469381666927, 3.88266032310549801636500608096, 4.90820479743024316726802631848, 6.95339861159347932366973503412, 7.60907036809425037459365469205, 8.060413050649067133210482798332, 9.618597627475434248324946749853, 9.861481985461325900707783042449, 10.76077115558820029057386114460

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