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Properties

Label 2-459-9.7-c1-0-14
Degree $2$
Conductor $459$
Sign $-0.809 + 0.586i$
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  + (0.974 − 1.68i)2-s + (−0.900 − 1.55i)4-s + (−0.974 − 1.68i)5-s + (0.900 − 1.55i)7-s + 0.389·8-s − 3.80·10-s + (0.194 − 0.337i)11-s + (−0.679 − 1.17i)13-s + (−1.75 − 3.03i)14-s + (2.17 − 3.77i)16-s + 17-s − 7.91·19-s + (−1.75 + 3.03i)20-s + (−0.379 − 0.657i)22-s + (0.905 + 1.56i)23-s + ⋯
L(s)  = 1  + (0.689 − 1.19i)2-s + (−0.450 − 0.779i)4-s + (−0.435 − 0.754i)5-s + (0.340 − 0.589i)7-s + 0.137·8-s − 1.20·10-s + (0.0587 − 0.101i)11-s + (−0.188 − 0.326i)13-s + (−0.468 − 0.812i)14-s + (0.544 − 0.943i)16-s + 0.242·17-s − 1.81·19-s + (−0.392 + 0.679i)20-s + (−0.0809 − 0.140i)22-s + (0.188 + 0.326i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(459\)    =    \(3^{3} \cdot 17\)
Sign: $-0.809 + 0.586i$
Analytic conductor: \(3.66513\)
Root analytic conductor: \(1.91445\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{459} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 459,\ (\ :1/2),\ -0.809 + 0.586i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.584630 - 1.80432i\)
\(L(\frac12)\) \(\approx\) \(0.584630 - 1.80432i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good2 \( 1 + (-0.974 + 1.68i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.974 + 1.68i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.900 + 1.55i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.194 + 0.337i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.679 + 1.17i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + 7.91T + 19T^{2} \)
23 \( 1 + (-0.905 - 1.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.26 - 3.93i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.40 - 4.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.20T + 37T^{2} \)
41 \( 1 + (-1.00 - 1.74i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.28 + 10.8i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.07 + 3.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.08T + 53T^{2} \)
59 \( 1 + (-5.80 - 10.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.74 - 8.21i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.81 - 3.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.5T + 71T^{2} \)
73 \( 1 - 0.111T + 73T^{2} \)
79 \( 1 + (5.36 - 9.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.10 - 3.64i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.18T + 89T^{2} \)
97 \( 1 + (-2.42 + 4.20i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.77584586814750750194941687846, −10.29716179211543987939170338125, −8.981626334136825788498366432233, −8.091088433233491220518188693316, −7.06676361855845962374300929912, −5.52450629528482420843120164346, −4.47413148250163163136288544454, −3.90166903066062545372918324129, −2.48662375311980226648159929474, −1.02622750530232039876315096333, 2.34334423402296457673357168216, 3.94337318674106459856057783989, 4.79516413746234643723416101105, 6.02205610529707992391418579426, 6.59388760321912604152845824681, 7.63411969610798432634401076502, 8.271594479019035312426195912096, 9.477632028753482322005214275376, 10.73750981466086457087899512181, 11.36156186327078599177270630402

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