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Properties

Label 2-460-115.102-c1-0-7
Degree $2$
Conductor $460$
Sign $0.954 + 0.298i$
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  + (1.31 − 0.285i)3-s + (2.00 − 0.995i)5-s + (1.10 − 0.605i)7-s + (−1.09 + 0.498i)9-s + (3.48 + 3.02i)11-s + (−0.652 + 1.19i)13-s + (2.34 − 1.87i)15-s + (−0.557 − 0.416i)17-s + (1.03 − 7.21i)19-s + (1.28 − 1.11i)21-s + (−4.78 − 0.276i)23-s + (3.01 − 3.98i)25-s + (−4.51 + 3.37i)27-s + (3.19 − 0.459i)29-s + (−7.47 + 4.80i)31-s + ⋯
L(s)  = 1  + (0.757 − 0.164i)3-s + (0.895 − 0.445i)5-s + (0.419 − 0.228i)7-s + (−0.363 + 0.166i)9-s + (1.05 + 0.911i)11-s + (−0.181 + 0.331i)13-s + (0.604 − 0.484i)15-s + (−0.135 − 0.101i)17-s + (0.238 − 1.65i)19-s + (0.279 − 0.242i)21-s + (−0.998 − 0.0576i)23-s + (0.603 − 0.797i)25-s + (−0.868 + 0.649i)27-s + (0.593 − 0.0853i)29-s + (−1.34 + 0.862i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.954 + 0.298i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.954 + 0.298i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06750 - 0.315758i\)
\(L(\frac12)\) \(\approx\) \(2.06750 - 0.315758i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.00 + 0.995i)T \)
23 \( 1 + (4.78 + 0.276i)T \)
good3 \( 1 + (-1.31 + 0.285i)T + (2.72 - 1.24i)T^{2} \)
7 \( 1 + (-1.10 + 0.605i)T + (3.78 - 5.88i)T^{2} \)
11 \( 1 + (-3.48 - 3.02i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (0.652 - 1.19i)T + (-7.02 - 10.9i)T^{2} \)
17 \( 1 + (0.557 + 0.416i)T + (4.78 + 16.3i)T^{2} \)
19 \( 1 + (-1.03 + 7.21i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-3.19 + 0.459i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (7.47 - 4.80i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-8.20 - 3.06i)T + (27.9 + 24.2i)T^{2} \)
41 \( 1 + (-1.02 + 2.23i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (0.116 + 0.536i)T + (-39.1 + 17.8i)T^{2} \)
47 \( 1 + (-0.150 - 0.150i)T + 47iT^{2} \)
53 \( 1 + (6.65 + 12.1i)T + (-28.6 + 44.5i)T^{2} \)
59 \( 1 + (-0.445 - 1.51i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (-5.29 - 8.24i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (4.64 + 0.332i)T + (66.3 + 9.53i)T^{2} \)
71 \( 1 + (-0.755 - 0.871i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (6.16 + 8.24i)T + (-20.5 + 70.0i)T^{2} \)
79 \( 1 + (16.2 - 4.77i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (4.25 - 11.3i)T + (-62.7 - 54.3i)T^{2} \)
89 \( 1 + (-9.15 - 5.88i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (-0.736 - 1.97i)T + (-73.3 + 63.5i)T^{2} \)
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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

−11.04646598629506851193818181222, −9.809377058728996027955518541379, −9.203840017815484409616806921569, −8.497258409746692024003337088176, −7.36257141302290946062118931370, −6.47171778896483567503073841210, −5.18281551133788618030869124762, −4.25303114542624818111054019760, −2.64551910720668039671716021334, −1.60946905936910569302430918900, 1.75956910314964108037900279780, 3.02244737404687908102691075872, 4.00806359764307931840583861508, 5.76519918600578618946037731299, 6.13104952417723309149020314360, 7.65113355871314623460117591369, 8.483158058102046691158735402365, 9.324180317522190476832501632457, 10.00095213796118019087226972880, 11.09584687835530095796273996008

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