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Properties

Label 2-495-5.2-c2-0-13
Degree $2$
Conductor $495$
Sign $-0.997 + 0.0695i$
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  + (2.40 + 2.40i)2-s + 7.55i·4-s + (−2.32 + 4.42i)5-s + (3.28 + 3.28i)7-s + (−8.54 + 8.54i)8-s + (−16.2 + 5.04i)10-s + 3.31·11-s + (−9.57 + 9.57i)13-s + 15.8i·14-s − 10.8·16-s + (−1.58 − 1.58i)17-s − 19.5i·19-s + (−33.4 − 17.5i)20-s + (7.97 + 7.97i)22-s + (−14.0 + 14.0i)23-s + ⋯
L(s)  = 1  + (1.20 + 1.20i)2-s + 1.88i·4-s + (−0.465 + 0.885i)5-s + (0.469 + 0.469i)7-s + (−1.06 + 1.06i)8-s + (−1.62 + 0.504i)10-s + 0.301·11-s + (−0.736 + 0.736i)13-s + 1.12i·14-s − 0.678·16-s + (−0.0934 − 0.0934i)17-s − 1.02i·19-s + (−1.67 − 0.878i)20-s + (0.362 + 0.362i)22-s + (−0.609 + 0.609i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0695i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.997 + 0.0695i$
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: \((2,\ 495,\ (\ :1),\ -0.997 + 0.0695i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.867166904\)
\(L(\frac12)\) \(\approx\) \(2.867166904\)
\(L(2)\) 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 \( 1 + (2.32 - 4.42i)T \)
11 \( 1 - 3.31T \)
good2 \( 1 + (-2.40 - 2.40i)T + 4iT^{2} \)
7 \( 1 + (-3.28 - 3.28i)T + 49iT^{2} \)
13 \( 1 + (9.57 - 9.57i)T - 169iT^{2} \)
17 \( 1 + (1.58 + 1.58i)T + 289iT^{2} \)
19 \( 1 + 19.5iT - 361T^{2} \)
23 \( 1 + (14.0 - 14.0i)T - 529iT^{2} \)
29 \( 1 - 30.9iT - 841T^{2} \)
31 \( 1 - 37.4T + 961T^{2} \)
37 \( 1 + (36.6 + 36.6i)T + 1.36e3iT^{2} \)
41 \( 1 + 6.31T + 1.68e3T^{2} \)
43 \( 1 + (1.92 - 1.92i)T - 1.84e3iT^{2} \)
47 \( 1 + (-40.9 - 40.9i)T + 2.20e3iT^{2} \)
53 \( 1 + (-22.1 + 22.1i)T - 2.80e3iT^{2} \)
59 \( 1 - 71.4iT - 3.48e3T^{2} \)
61 \( 1 - 83.5T + 3.72e3T^{2} \)
67 \( 1 + (12.3 + 12.3i)T + 4.48e3iT^{2} \)
71 \( 1 - 46.1T + 5.04e3T^{2} \)
73 \( 1 + (43.0 - 43.0i)T - 5.32e3iT^{2} \)
79 \( 1 - 95.5iT - 6.24e3T^{2} \)
83 \( 1 + (-91.7 + 91.7i)T - 6.88e3iT^{2} \)
89 \( 1 - 133. iT - 7.92e3T^{2} \)
97 \( 1 + (-39.9 - 39.9i)T + 9.40e3iT^{2} \)
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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

−11.65214164445165977435273220919, −10.45051642622344916377427640650, −9.122553393467198240818205724600, −8.100147247934507554438269476451, −7.15014626535961117578398049989, −6.71291274242504804287956621267, −5.57435945675130841411501005488, −4.65622489313107526389344000594, −3.71966061729062991331000272061, −2.48525764394324933907127862045, 0.790684088281053075966258551683, 2.07538766917955964518439869591, 3.50554135058843775352940943925, 4.37844244475228622490294585772, 5.06047646995314979174969204981, 6.10862420331109632560291356787, 7.67259787238867582505846776684, 8.488099551048115733455845168508, 9.920390176324464027684775090200, 10.38821850883376427186544108149

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