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Properties

Label 2-459-153.106-c1-0-6
Degree $2$
Conductor $459$
Sign $0.929 + 0.369i$
Analytic cond. $3.66513$
Root an. cond. $1.91445$
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  + (−0.218 − 0.126i)2-s + (−0.968 − 1.67i)4-s + (3.83 + 1.02i)5-s + (2.05 − 0.551i)7-s + 0.995i·8-s + (−0.709 − 0.709i)10-s + (−1.14 + 0.307i)11-s + (3.09 + 5.36i)13-s + (−0.520 − 0.139i)14-s + (−1.81 + 3.13i)16-s + (−2.45 − 3.31i)17-s − 5.66i·19-s + (−1.98 − 7.41i)20-s + (0.290 + 0.0778i)22-s + (0.188 − 0.703i)23-s + ⋯
L(s)  = 1  + (−0.154 − 0.0893i)2-s + (−0.484 − 0.838i)4-s + (1.71 + 0.459i)5-s + (0.778 − 0.208i)7-s + 0.351i·8-s + (−0.224 − 0.224i)10-s + (−0.346 + 0.0928i)11-s + (0.858 + 1.48i)13-s + (−0.139 − 0.0372i)14-s + (−0.452 + 0.783i)16-s + (−0.595 − 0.803i)17-s − 1.29i·19-s + (−0.444 − 1.65i)20-s + (0.0619 + 0.0165i)22-s + (0.0392 − 0.146i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(459\)    =    \(3^{3} \cdot 17\)
Sign: $0.929 + 0.369i$
Analytic conductor: \(3.66513\)
Root analytic conductor: \(1.91445\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{459} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 459,\ (\ :1/2),\ 0.929 + 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62156 - 0.310995i\)
\(L(\frac12)\) \(\approx\) \(1.62156 - 0.310995i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 + (2.45 + 3.31i)T \)
good2 \( 1 + (0.218 + 0.126i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-3.83 - 1.02i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-2.05 + 0.551i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.14 - 0.307i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.09 - 5.36i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + 5.66iT - 19T^{2} \)
23 \( 1 + (-0.188 + 0.703i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.799 - 2.98i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-0.0434 - 0.0116i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.68 + 1.68i)T - 37iT^{2} \)
41 \( 1 + (-0.174 + 0.650i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.442 + 0.255i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.74 + 8.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.10iT - 53T^{2} \)
59 \( 1 + (7.91 - 4.56i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.59 - 1.49i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.68 + 2.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.32 - 2.32i)T - 71iT^{2} \)
73 \( 1 + (8.24 - 8.24i)T - 73iT^{2} \)
79 \( 1 + (0.124 - 0.0332i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (5.37 + 3.10i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.09T + 89T^{2} \)
97 \( 1 + (-1.05 - 3.94i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.89187803746652319884107200636, −10.10979252240225599494109411965, −9.165210863860440807267835926471, −8.827718459734963039858623891937, −7.03690286083764834488021018702, −6.30728772968983756598432663810, −5.28779307087957943832397256169, −4.51083958568445782618484724248, −2.42712187370288480166860743304, −1.46834122641862635276485038944, 1.52047536682668799647628589153, 2.94785054804758133160795683632, 4.43043738345262094201423221727, 5.55916425024461958268148159801, 6.16356750013453104279419128950, 7.84948267549001615392883385137, 8.373883994935182036573603137016, 9.219087528855299942006236687120, 10.15614619939716211817580812035, 10.88197611327407949004402447276

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