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Properties

Label 2-459-153.101-c2-0-26
Degree $2$
Conductor $459$
Sign $0.221 - 0.975i$
Analytic cond. $12.5068$
Root an. cond. $3.53650$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.19 − 1.84i)2-s + (4.80 + 8.32i)4-s + (−2.19 − 3.79i)5-s + (−6.88 − 3.97i)7-s − 20.7i·8-s + 16.1i·10-s + (−1.03 + 1.80i)11-s + (−9.18 − 15.9i)13-s + (14.6 + 25.4i)14-s + (−18.9 + 32.8i)16-s + (9.71 − 13.9i)17-s + 21.4·19-s + (21.0 − 36.5i)20-s + (6.64 − 3.83i)22-s + (−10.9 − 18.9i)23-s + ⋯
L(s)  = 1  + (−1.59 − 0.922i)2-s + (1.20 + 2.08i)4-s + (−0.438 − 0.759i)5-s + (−0.983 − 0.568i)7-s − 2.58i·8-s + 1.61i·10-s + (−0.0944 + 0.163i)11-s + (−0.706 − 1.22i)13-s + (1.04 + 1.81i)14-s + (−1.18 + 2.05i)16-s + (0.571 − 0.820i)17-s + 1.12·19-s + (1.05 − 1.82i)20-s + (0.301 − 0.174i)22-s + (−0.476 − 0.824i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(459\)    =    \(3^{3} \cdot 17\)
Sign: $0.221 - 0.975i$
Analytic conductor: \(12.5068\)
Root analytic conductor: \(3.53650\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{459} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 459,\ (\ :1),\ 0.221 - 0.975i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.06864662990\)
\(L(\frac12)\) \(\approx\) \(0.06864662990\)
\(L(2)\) 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 \( 1 + (-9.71 + 13.9i)T \)
good2 \( 1 + (3.19 + 1.84i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (2.19 + 3.79i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (6.88 + 3.97i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (1.03 - 1.80i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (9.18 + 15.9i)T + (-84.5 + 146. i)T^{2} \)
19 \( 1 - 21.4T + 361T^{2} \)
23 \( 1 + (10.9 + 18.9i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (16.2 - 28.1i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (41.8 - 24.1i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 24.0iT - 1.36e3T^{2} \)
41 \( 1 + (17.7 + 30.7i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (28.7 - 49.7i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-30.6 - 17.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 32.6iT - 2.80e3T^{2} \)
59 \( 1 + (-30.3 + 17.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-40.3 - 23.2i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-13.0 - 22.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 119.T + 5.04e3T^{2} \)
73 \( 1 - 106. iT - 5.32e3T^{2} \)
79 \( 1 + (-74.5 - 43.0i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-5.09 - 2.93i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 83.7iT - 7.92e3T^{2} \)
97 \( 1 + (89.0 + 51.4i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.05467686653346254541227647035, −9.420759966651432467858466780326, −8.552968917966242036913116361963, −7.58941405644190196637491487806, −7.07219275636117985113578050882, −5.27266245087584790759714111831, −3.64989137412914814228267494007, −2.76289443301078060175355516823, −0.999588401979641863103219137690, −0.05907847308973359425189015148, 1.93604762480539888806368730788, 3.47915236409472169461850135400, 5.51171708492121191991991275089, 6.33540322806325511294780456520, 7.20147461668248008395697609205, 7.76626435366815280893192238608, 8.953593198736168352760542154308, 9.618564470363864932480930203482, 10.19195197004985281023232590964, 11.34796012032074471154302223580

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