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Properties

Label 2-495-5.2-c2-0-35
Degree $2$
Conductor $495$
Sign $0.829 + 0.558i$
Analytic cond. $13.4877$
Root an. cond. $3.67257$
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  + (0.692 + 0.692i)2-s − 3.03i·4-s + (4.55 − 2.05i)5-s + (3.68 + 3.68i)7-s + (4.87 − 4.87i)8-s + (4.58 + 1.73i)10-s + 3.31·11-s + (8.16 − 8.16i)13-s + 5.10i·14-s − 5.39·16-s + (−8.96 − 8.96i)17-s + 15.2i·19-s + (−6.25 − 13.8i)20-s + (2.29 + 2.29i)22-s + (−15.0 + 15.0i)23-s + ⋯
L(s)  = 1  + (0.346 + 0.346i)2-s − 0.759i·4-s + (0.911 − 0.411i)5-s + (0.525 + 0.525i)7-s + (0.609 − 0.609i)8-s + (0.458 + 0.173i)10-s + 0.301·11-s + (0.628 − 0.628i)13-s + 0.364i·14-s − 0.337·16-s + (−0.527 − 0.527i)17-s + 0.803i·19-s + (−0.312 − 0.692i)20-s + (0.104 + 0.104i)22-s + (−0.656 + 0.656i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.829 + 0.558i$
Analytic conductor: \(13.4877\)
Root analytic conductor: \(3.67257\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1),\ 0.829 + 0.558i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.715865005\)
\(L(\frac12)\) \(\approx\) \(2.715865005\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-4.55 + 2.05i)T \)
11 \( 1 - 3.31T \)
good2 \( 1 + (-0.692 - 0.692i)T + 4iT^{2} \)
7 \( 1 + (-3.68 - 3.68i)T + 49iT^{2} \)
13 \( 1 + (-8.16 + 8.16i)T - 169iT^{2} \)
17 \( 1 + (8.96 + 8.96i)T + 289iT^{2} \)
19 \( 1 - 15.2iT - 361T^{2} \)
23 \( 1 + (15.0 - 15.0i)T - 529iT^{2} \)
29 \( 1 + 17.7iT - 841T^{2} \)
31 \( 1 - 49.7T + 961T^{2} \)
37 \( 1 + (26.1 + 26.1i)T + 1.36e3iT^{2} \)
41 \( 1 + 12.6T + 1.68e3T^{2} \)
43 \( 1 + (-30.0 + 30.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (-34.2 - 34.2i)T + 2.20e3iT^{2} \)
53 \( 1 + (55.2 - 55.2i)T - 2.80e3iT^{2} \)
59 \( 1 + 57.6iT - 3.48e3T^{2} \)
61 \( 1 - 96.1T + 3.72e3T^{2} \)
67 \( 1 + (-64.4 - 64.4i)T + 4.48e3iT^{2} \)
71 \( 1 + 48.5T + 5.04e3T^{2} \)
73 \( 1 + (59.7 - 59.7i)T - 5.32e3iT^{2} \)
79 \( 1 + 27.3iT - 6.24e3T^{2} \)
83 \( 1 + (-9.39 + 9.39i)T - 6.88e3iT^{2} \)
89 \( 1 - 120. iT - 7.92e3T^{2} \)
97 \( 1 + (-35.0 - 35.0i)T + 9.40e3iT^{2} \)
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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

−10.50156120273358463034074143914, −9.797545211653633230550383807950, −8.945130552943332612543840006626, −7.993544716114449744231703185696, −6.64203142487158628976981008267, −5.81566857266710026113464232503, −5.25466135827267119438362479150, −4.12536636009550353375829997464, −2.27451085498848924070077508423, −1.11482735724741201192948649600, 1.62853075214582860738509924294, 2.76776824598249623518976281411, 4.02042038353184561789658615140, 4.86971235262427804413146100023, 6.32531606429964510864230082938, 7.01913479165470228529290915271, 8.236272644759229508156902726752, 8.943660157918497121815370838159, 10.14962274421791037795313515474, 10.94812744388232781280214561234

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