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Properties

Label 2-495-11.9-c1-0-10
Degree $2$
Conductor $495$
Sign $0.921 - 0.388i$
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  + (−0.697 + 2.14i)2-s + (−2.50 − 1.82i)4-s + (−0.309 − 0.951i)5-s + (−0.100 − 0.0728i)7-s + (2.00 − 1.45i)8-s + 2.25·10-s + (0.899 − 3.19i)11-s + (1.69 − 5.22i)13-s + (0.226 − 0.164i)14-s + (−0.184 − 0.566i)16-s + (−0.160 − 0.494i)17-s + (−2.55 + 1.85i)19-s + (−0.957 + 2.94i)20-s + (6.22 + 4.15i)22-s + 7.92·23-s + ⋯
L(s)  = 1  + (−0.493 + 1.51i)2-s + (−1.25 − 0.910i)4-s + (−0.138 − 0.425i)5-s + (−0.0379 − 0.0275i)7-s + (0.709 − 0.515i)8-s + 0.714·10-s + (0.271 − 0.962i)11-s + (0.470 − 1.44i)13-s + (0.0605 − 0.0439i)14-s + (−0.0460 − 0.141i)16-s + (−0.0389 − 0.119i)17-s + (−0.586 + 0.426i)19-s + (−0.214 + 0.658i)20-s + (1.32 + 0.886i)22-s + 1.65·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.921 - 0.388i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.921 - 0.388i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.881792 + 0.178456i\)
\(L(\frac12)\) \(\approx\) \(0.881792 + 0.178456i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.899 + 3.19i)T \)
good2 \( 1 + (0.697 - 2.14i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (0.100 + 0.0728i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.69 + 5.22i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.160 + 0.494i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.55 - 1.85i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 7.92T + 23T^{2} \)
29 \( 1 + (3.29 + 2.39i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.17 + 6.70i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-7.10 - 5.16i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (6.10 - 4.43i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.42T + 43T^{2} \)
47 \( 1 + (-0.369 + 0.268i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.0109 - 0.0337i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (4.42 + 3.21i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.37 - 7.31i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 2.53T + 67T^{2} \)
71 \( 1 + (3.79 + 11.6i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.89 + 5.00i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.93 + 5.96i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.193 + 0.595i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 10.1T + 89T^{2} \)
97 \( 1 + (0.567 - 1.74i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.85453123667645621059712352749, −9.768194133364173068106238908829, −8.843837865308213660950162754552, −8.215228610821711296030109359280, −7.53732868380419274286762331808, −6.31555966107227621090523248348, −5.74160403680728454306904929113, −4.70328506674920145742404698821, −3.19200570804083067385546548662, −0.69942163222639236284108520211, 1.48667697873182869157171774834, 2.59781971091851924441181336374, 3.79966107083147071871744925328, 4.69699435305986426476824678844, 6.47877076672658911766098738167, 7.28087835205576182820882850317, 8.800936236108523930368101424058, 9.185155401598697943490060625456, 10.16181380984941676384115753658, 11.02220019402585040936078886102

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