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Properties

Label 2-460-115.104-c1-0-5
Degree $2$
Conductor $460$
Sign $0.381 - 0.924i$
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  + (3.24 + 0.466i)3-s + (−0.926 + 2.03i)5-s + (−3.45 + 2.99i)7-s + (7.42 + 2.17i)9-s + (−2.75 − 1.76i)11-s + (1.02 + 0.889i)13-s + (−3.95 + 6.16i)15-s + (4.62 − 2.11i)17-s + (−0.171 + 0.374i)19-s + (−12.5 + 8.09i)21-s + (4.78 − 0.363i)23-s + (−3.28 − 3.76i)25-s + (14.1 + 6.44i)27-s + (1.72 + 3.78i)29-s + (−0.905 − 6.29i)31-s + ⋯
L(s)  = 1  + (1.87 + 0.269i)3-s + (−0.414 + 0.910i)5-s + (−1.30 + 1.13i)7-s + (2.47 + 0.726i)9-s + (−0.830 − 0.533i)11-s + (0.284 + 0.246i)13-s + (−1.02 + 1.59i)15-s + (1.12 − 0.512i)17-s + (−0.0392 + 0.0859i)19-s + (−2.74 + 1.76i)21-s + (0.997 − 0.0757i)23-s + (−0.656 − 0.753i)25-s + (2.71 + 1.24i)27-s + (0.321 + 0.703i)29-s + (−0.162 − 1.13i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.381 - 0.924i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.381 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78503 + 1.19398i\)
\(L(\frac12)\) \(\approx\) \(1.78503 + 1.19398i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.926 - 2.03i)T \)
23 \( 1 + (-4.78 + 0.363i)T \)
good3 \( 1 + (-3.24 - 0.466i)T + (2.87 + 0.845i)T^{2} \)
7 \( 1 + (3.45 - 2.99i)T + (0.996 - 6.92i)T^{2} \)
11 \( 1 + (2.75 + 1.76i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (-1.02 - 0.889i)T + (1.85 + 12.8i)T^{2} \)
17 \( 1 + (-4.62 + 2.11i)T + (11.1 - 12.8i)T^{2} \)
19 \( 1 + (0.171 - 0.374i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-1.72 - 3.78i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (0.905 + 6.29i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (-3.21 + 10.9i)T + (-31.1 - 20.0i)T^{2} \)
41 \( 1 + (1.44 - 0.424i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-0.0389 - 0.00560i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + 2.24iT - 47T^{2} \)
53 \( 1 + (3.04 - 2.64i)T + (7.54 - 52.4i)T^{2} \)
59 \( 1 + (-0.959 + 1.10i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (-0.685 - 4.76i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (-0.149 - 0.232i)T + (-27.8 + 60.9i)T^{2} \)
71 \( 1 + (10.4 - 6.70i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-10.5 - 4.83i)T + (47.8 + 55.1i)T^{2} \)
79 \( 1 + (1.78 - 2.05i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (1.18 - 4.02i)T + (-69.8 - 44.8i)T^{2} \)
89 \( 1 + (0.786 - 5.47i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-0.0232 - 0.0791i)T + (-81.6 + 52.4i)T^{2} \)
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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

−11.00840735333148411309418491598, −10.00859209491411516359278141175, −9.381012908816217610893500547740, −8.600322058324752810580624024852, −7.71961906900639906276464612637, −6.89385689586792415707774663640, −5.60434417912636368276073645587, −3.85123903588269243870769370636, −3.00852615859893887929164873528, −2.54610222664788868209397160463, 1.22180176136024031798775609380, 3.00338373664417075273371868309, 3.65894830880157910736425550016, 4.75818972013894438054139363384, 6.60324487463104236497563328171, 7.57879985800166931209297011904, 8.064386478651381856473779131424, 9.045535276719585978331539192057, 9.845812348417587718778859935132, 10.40514603232904662083056015220

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