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Properties

Label 2-495-11.5-c1-0-12
Degree $2$
Conductor $495$
Sign $-0.999 - 0.0101i$
Analytic cond. $3.95259$
Root an. cond. $1.98811$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.788 − 2.42i)2-s + (−3.64 + 2.65i)4-s + (0.309 − 0.951i)5-s + (3.39 − 2.46i)7-s + (5.18 + 3.76i)8-s − 2.55·10-s + (2.04 + 2.61i)11-s + (−1.52 − 4.69i)13-s + (−8.65 − 6.29i)14-s + (2.26 − 6.96i)16-s + (1.88 − 5.80i)17-s + (2.03 + 1.47i)19-s + (1.39 + 4.28i)20-s + (4.72 − 7.01i)22-s − 3.08·23-s + ⋯
L(s)  = 1  + (−0.557 − 1.71i)2-s + (−1.82 + 1.32i)4-s + (0.138 − 0.425i)5-s + (1.28 − 0.931i)7-s + (1.83 + 1.33i)8-s − 0.806·10-s + (0.616 + 0.787i)11-s + (−0.422 − 1.30i)13-s + (−2.31 − 1.68i)14-s + (0.565 − 1.74i)16-s + (0.457 − 1.40i)17-s + (0.465 + 0.338i)19-s + (0.311 + 0.959i)20-s + (1.00 − 1.49i)22-s − 0.643·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0101i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.999 - 0.0101i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.999 - 0.0101i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00542241 + 1.06702i\)
\(L(\frac12)\) \(\approx\) \(0.00542241 + 1.06702i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-2.04 - 2.61i)T \)
good2 \( 1 + (0.788 + 2.42i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (-3.39 + 2.46i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.52 + 4.69i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.88 + 5.80i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.03 - 1.47i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.08T + 23T^{2} \)
29 \( 1 + (2.91 - 2.11i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.616 + 1.89i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (6.18 - 4.49i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (2.39 + 1.73i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 0.186T + 43T^{2} \)
47 \( 1 + (-4.50 - 3.27i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.92 - 5.91i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.429 - 0.311i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.51 + 7.75i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 6.64T + 67T^{2} \)
71 \( 1 + (-3.40 + 10.4i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.0361 - 0.0262i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.87 - 11.9i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.99 + 6.15i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 0.535T + 89T^{2} \)
97 \( 1 + (3.41 + 10.5i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.44219857047427225718170315535, −9.904571170933600881149132534571, −9.045754125801475257661810297519, −7.965322957199271188420607966384, −7.39978442199668927133858287152, −5.22137855150645302990428710257, −4.46482602893316131620616247060, −3.34318020522954078711427735897, −1.91508442905222973308036018387, −0.857944538188016119850375289328, 1.78681515686238012040615522687, 4.05490175091307024709467566771, 5.27254933137002410203869246202, 5.95252057606750711422688194306, 6.84769438970860976463907515528, 7.79825128906762857526723950996, 8.617650604934019769208266061108, 9.099046909216886128440715109104, 10.19585205450083572772233339436, 11.34224455271454144374596430591

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