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Properties

Label 2-495-45.4-c1-0-56
Degree $2$
Conductor $495$
Sign $-0.159 + 0.987i$
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  + (2.02 − 1.16i)2-s + (0.597 − 1.62i)3-s + (1.72 − 2.99i)4-s + (1.74 + 1.40i)5-s + (−0.690 − 3.98i)6-s + (−1.18 + 0.684i)7-s − 3.39i·8-s + (−2.28 − 1.94i)9-s + (5.15 + 0.803i)10-s + (−0.5 − 0.866i)11-s + (−3.83 − 4.59i)12-s + (−0.842 − 0.486i)13-s + (−1.59 + 2.76i)14-s + (3.32 − 1.99i)15-s + (−0.507 − 0.879i)16-s + 0.509i·17-s + ⋯
L(s)  = 1  + (1.42 − 0.825i)2-s + (0.344 − 0.938i)3-s + (0.863 − 1.49i)4-s + (0.778 + 0.627i)5-s + (−0.281 − 1.62i)6-s + (−0.448 + 0.258i)7-s − 1.19i·8-s + (−0.762 − 0.647i)9-s + (1.63 + 0.254i)10-s + (−0.150 − 0.261i)11-s + (−1.10 − 1.32i)12-s + (−0.233 − 0.134i)13-s + (−0.427 + 0.740i)14-s + (0.857 − 0.514i)15-s + (−0.126 − 0.219i)16-s + 0.123i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.159 + 0.987i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.159 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.24241 - 2.63474i\)
\(L(\frac12)\) \(\approx\) \(2.24241 - 2.63474i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.597 + 1.62i)T \)
5 \( 1 + (-1.74 - 1.40i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-2.02 + 1.16i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (1.18 - 0.684i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (0.842 + 0.486i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.509iT - 17T^{2} \)
19 \( 1 - 2.24T + 19T^{2} \)
23 \( 1 + (3.88 + 2.24i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.63 - 4.56i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.491 - 0.852i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.38iT - 37T^{2} \)
41 \( 1 + (0.164 - 0.285i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.53 + 2.61i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.45 - 3.72i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.40iT - 53T^{2} \)
59 \( 1 + (3.01 - 5.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.08 + 3.60i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.72 + 5.61i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + 14.7iT - 73T^{2} \)
79 \( 1 + (8.77 + 15.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.37 + 1.37i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.93T + 89T^{2} \)
97 \( 1 + (13.0 - 7.50i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.94846242375498856584870226011, −10.13678880796678955208585379078, −9.082972147844463701905012513282, −7.79216618443048605915829210118, −6.52832727729858017939900377191, −6.05443059161689530081379818799, −5.00926773298539151695842461347, −3.38529022671384093790686427180, −2.75732381042868432513805602999, −1.68902150554426894263255159535, 2.52907247937735964939861080372, 3.76680577065437126477017329624, 4.60366826364212344729859087807, 5.44068354721046794563004834417, 6.16151508426712825341084350352, 7.36429256394817558112900712346, 8.377335922095119384054045330171, 9.588764175961255098109812547561, 10.02043541070723447622184035948, 11.39900122474262236429864582711

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