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Properties

Label 2-495-11.9-c1-0-12
Degree $2$
Conductor $495$
Sign $0.907 + 0.420i$
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  + (−0.758 + 2.33i)2-s + (−3.26 − 2.36i)4-s + (0.309 + 0.951i)5-s + (−2.65 − 1.93i)7-s + (4.03 − 2.93i)8-s − 2.45·10-s + (−2.96 + 1.47i)11-s + (−0.0967 + 0.297i)13-s + (6.53 − 4.74i)14-s + (1.29 + 3.98i)16-s + (−1.54 − 4.75i)17-s + (6.03 − 4.38i)19-s + (1.24 − 3.83i)20-s + (−1.20 − 8.05i)22-s − 1.07·23-s + ⋯
L(s)  = 1  + (−0.536 + 1.65i)2-s + (−1.63 − 1.18i)4-s + (0.138 + 0.425i)5-s + (−1.00 − 0.730i)7-s + (1.42 − 1.03i)8-s − 0.776·10-s + (−0.894 + 0.446i)11-s + (−0.0268 + 0.0825i)13-s + (1.74 − 1.26i)14-s + (0.323 + 0.996i)16-s + (−0.374 − 1.15i)17-s + (1.38 − 1.00i)19-s + (0.278 − 0.857i)20-s + (−0.256 − 1.71i)22-s − 0.223·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.907 + 0.420i$
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: \((2,\ 495,\ (\ :1/2),\ 0.907 + 0.420i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.336660 - 0.0742564i\)
\(L(\frac12)\) \(\approx\) \(0.336660 - 0.0742564i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.96 - 1.47i)T \)
good2 \( 1 + (0.758 - 2.33i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (2.65 + 1.93i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.0967 - 0.297i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.54 + 4.75i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-6.03 + 4.38i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 1.07T + 23T^{2} \)
29 \( 1 + (4.07 + 2.96i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.06 + 3.28i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.13 + 1.54i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-8.77 + 6.37i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.51T + 43T^{2} \)
47 \( 1 + (9.70 - 7.05i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.52 - 4.69i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (7.41 + 5.38i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.83 - 8.73i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 15.2T + 67T^{2} \)
71 \( 1 + (0.949 + 2.92i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.00 - 5.08i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.67 + 5.14i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (5.02 + 15.4i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 1.62T + 89T^{2} \)
97 \( 1 + (-0.0692 + 0.213i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.46876612343967702166429871568, −9.597841209284942303945088985070, −9.210146317268264534539640118659, −7.68265337050251486423790562363, −7.33773180367758567946047741202, −6.53416967562344677613945116654, −5.55330368697993394189419999196, −4.55040782769677690848656899267, −2.93201043973810625499773282450, −0.24695823321067149268394806236, 1.58828431791940503924612498044, 2.90303410377033029125188035489, 3.65637003258953638332978896279, 5.16658977531223792311105489835, 6.21207972566412606547255365704, 7.921593521093208198154896488186, 8.644713270529956696008922446293, 9.545343112667243138998186850518, 10.07261455485149510100728529896, 10.94416670070961213722796304416

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