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Properties

Label 2-459-153.101-c2-0-18
Degree $2$
Conductor $459$
Sign $0.865 - 0.500i$
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  + (1.95 + 1.12i)2-s + (0.545 + 0.944i)4-s + (−0.723 − 1.25i)5-s + (9.69 + 5.59i)7-s − 6.56i·8-s − 3.26i·10-s + (−2.23 + 3.87i)11-s + (2.00 + 3.47i)13-s + (12.6 + 21.8i)14-s + (9.58 − 16.6i)16-s + (10.2 − 13.5i)17-s + 10.9·19-s + (0.789 − 1.36i)20-s + (−8.73 + 5.04i)22-s + (19.8 + 34.4i)23-s + ⋯
L(s)  = 1  + (0.976 + 0.564i)2-s + (0.136 + 0.236i)4-s + (−0.144 − 0.250i)5-s + (1.38 + 0.799i)7-s − 0.820i·8-s − 0.326i·10-s + (−0.203 + 0.352i)11-s + (0.154 + 0.267i)13-s + (0.901 + 1.56i)14-s + (0.599 − 1.03i)16-s + (0.602 − 0.797i)17-s + 0.578·19-s + (0.0394 − 0.0683i)20-s + (−0.397 + 0.229i)22-s + (0.864 + 1.49i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(459\)    =    \(3^{3} \cdot 17\)
Sign: $0.865 - 0.500i$
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: \((2,\ 459,\ (\ :1),\ 0.865 - 0.500i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.229095329\)
\(L(\frac12)\) \(\approx\) \(3.229095329\)
\(L(2)\) 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 \( 1 + (-10.2 + 13.5i)T \)
good2 \( 1 + (-1.95 - 1.12i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (0.723 + 1.25i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-9.69 - 5.59i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (2.23 - 3.87i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-2.00 - 3.47i)T + (-84.5 + 146. i)T^{2} \)
19 \( 1 - 10.9T + 361T^{2} \)
23 \( 1 + (-19.8 - 34.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-0.551 + 0.954i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-10.1 + 5.87i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 8.93iT - 1.36e3T^{2} \)
41 \( 1 + (6.83 + 11.8i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (39.5 - 68.4i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-46.1 - 26.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 72.6iT - 2.80e3T^{2} \)
59 \( 1 + (36.9 - 21.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (70.5 + 40.7i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (32.0 + 55.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 10.3T + 5.04e3T^{2} \)
73 \( 1 - 56.3iT - 5.32e3T^{2} \)
79 \( 1 + (33.6 + 19.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (110. + 63.8i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 79.8iT - 7.92e3T^{2} \)
97 \( 1 + (140. + 81.1i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.28882142983637737398000899197, −9.931681574944919024475331847953, −9.070988884478068481726611317097, −7.967661577469129214216277082062, −7.19182042055857851449989417155, −5.91882657663236163787336545387, −5.07456742960545262783367987041, −4.58491004580163680597789610326, −3.08183411771987114516172509640, −1.36413060230087441532508030276, 1.34088605073697899687011251478, 2.85611576007754894238400543359, 3.91827039783722594909924738221, 4.81161470058757518331472398181, 5.64558888995421008182093068985, 7.13896429851429450021296429661, 8.055032374300111638023443613543, 8.760958217476335075841372271630, 10.60876384491262020116283554668, 10.71610395751900414511029058320

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