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Properties

Label 2-495-5.2-c2-0-42
Degree $2$
Conductor $495$
Sign $0.932 + 0.359i$
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  + (1.89 + 1.89i)2-s + 3.15i·4-s + (3.98 − 3.02i)5-s + (−9.02 − 9.02i)7-s + (1.59 − 1.59i)8-s + (13.2 + 1.81i)10-s + 3.31·11-s + (−9.63 + 9.63i)13-s − 34.1i·14-s + 18.6·16-s + (2.76 + 2.76i)17-s − 34.0i·19-s + (9.53 + 12.5i)20-s + (6.27 + 6.27i)22-s + (20.6 − 20.6i)23-s + ⋯
L(s)  = 1  + (0.945 + 0.945i)2-s + 0.788i·4-s + (0.796 − 0.604i)5-s + (−1.28 − 1.28i)7-s + (0.199 − 0.199i)8-s + (1.32 + 0.181i)10-s + 0.301·11-s + (−0.740 + 0.740i)13-s − 2.43i·14-s + 1.16·16-s + (0.162 + 0.162i)17-s − 1.79i·19-s + (0.476 + 0.628i)20-s + (0.285 + 0.285i)22-s + (0.898 − 0.898i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.359i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.932 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.932 + 0.359i$
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: \((2,\ 495,\ (\ :1),\ 0.932 + 0.359i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.818178885\)
\(L(\frac12)\) \(\approx\) \(2.818178885\)
\(L(2)\) 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 \( 1 + (-3.98 + 3.02i)T \)
11 \( 1 - 3.31T \)
good2 \( 1 + (-1.89 - 1.89i)T + 4iT^{2} \)
7 \( 1 + (9.02 + 9.02i)T + 49iT^{2} \)
13 \( 1 + (9.63 - 9.63i)T - 169iT^{2} \)
17 \( 1 + (-2.76 - 2.76i)T + 289iT^{2} \)
19 \( 1 + 34.0iT - 361T^{2} \)
23 \( 1 + (-20.6 + 20.6i)T - 529iT^{2} \)
29 \( 1 + 23.7iT - 841T^{2} \)
31 \( 1 - 27.0T + 961T^{2} \)
37 \( 1 + (-37.1 - 37.1i)T + 1.36e3iT^{2} \)
41 \( 1 + 43.6T + 1.68e3T^{2} \)
43 \( 1 + (20.6 - 20.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (7.41 + 7.41i)T + 2.20e3iT^{2} \)
53 \( 1 + (36.6 - 36.6i)T - 2.80e3iT^{2} \)
59 \( 1 - 102. iT - 3.48e3T^{2} \)
61 \( 1 + 16.6T + 3.72e3T^{2} \)
67 \( 1 + (-10.5 - 10.5i)T + 4.48e3iT^{2} \)
71 \( 1 - 17.2T + 5.04e3T^{2} \)
73 \( 1 + (-8.68 + 8.68i)T - 5.32e3iT^{2} \)
79 \( 1 + 105. iT - 6.24e3T^{2} \)
83 \( 1 + (5.74 - 5.74i)T - 6.88e3iT^{2} \)
89 \( 1 + 172. iT - 7.92e3T^{2} \)
97 \( 1 + (-75.1 - 75.1i)T + 9.40e3iT^{2} \)
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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

−10.42445629679086516116307777129, −9.803051853397057114092667777535, −8.965982262335751593168056793395, −7.51133130911978040708713533403, −6.60780017671493913339005554721, −6.33563396965400832977488823754, −4.82254302428577049044573804669, −4.38317199601439369869487454398, −2.92631031125860881887000751285, −0.847931243955902893408765597522, 1.88024743214887500780138623148, 2.94521546565769775681995742125, 3.47382237409316073581003284430, 5.24277035567119230037273560389, 5.77311382692753555833137813691, 6.79435692025939089357064806219, 8.158920372299737413069374519625, 9.547942782653339963715561667059, 9.905619064005710551318462305213, 10.90025156270513932417541804939

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