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Properties

Label 2-495-55.52-c1-0-20
Degree $2$
Conductor $495$
Sign $-0.229 + 0.973i$
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  + (0.0763 − 0.482i)2-s + (1.67 + 0.544i)4-s + (−1.47 − 1.67i)5-s + (−1.32 − 2.59i)7-s + (0.833 − 1.63i)8-s + (−0.922 + 0.583i)10-s + (3.15 − 1.00i)11-s + (−2.89 − 0.457i)13-s + (−1.35 + 0.440i)14-s + (2.12 + 1.54i)16-s + (−5.20 + 0.824i)17-s + (−1.26 − 3.89i)19-s + (−1.55 − 3.61i)20-s + (−0.244 − 1.60i)22-s + (2.12 − 2.12i)23-s + ⋯
L(s)  = 1  + (0.0540 − 0.341i)2-s + (0.837 + 0.272i)4-s + (−0.660 − 0.750i)5-s + (−0.500 − 0.982i)7-s + (0.294 − 0.578i)8-s + (−0.291 + 0.184i)10-s + (0.952 − 0.303i)11-s + (−0.801 − 0.126i)13-s + (−0.362 + 0.117i)14-s + (0.531 + 0.385i)16-s + (−1.26 + 0.199i)17-s + (−0.290 − 0.894i)19-s + (−0.348 − 0.808i)20-s + (−0.0521 − 0.341i)22-s + (0.442 − 0.442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.229 + 0.973i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.229 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.875550 - 1.10629i\)
\(L(\frac12)\) \(\approx\) \(0.875550 - 1.10629i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.47 + 1.67i)T \)
11 \( 1 + (-3.15 + 1.00i)T \)
good2 \( 1 + (-0.0763 + 0.482i)T + (-1.90 - 0.618i)T^{2} \)
7 \( 1 + (1.32 + 2.59i)T + (-4.11 + 5.66i)T^{2} \)
13 \( 1 + (2.89 + 0.457i)T + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (5.20 - 0.824i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (1.26 + 3.89i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.12 + 2.12i)T - 23iT^{2} \)
29 \( 1 + (-0.817 + 2.51i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.45 + 3.96i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.03 + 0.528i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (-3.38 + 1.09i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (-5.07 - 5.07i)T + 43iT^{2} \)
47 \( 1 + (1.67 - 3.28i)T + (-27.6 - 38.0i)T^{2} \)
53 \( 1 + (-0.231 + 1.45i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (1.52 + 0.496i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.07 - 5.61i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.31 - 1.31i)T + 67iT^{2} \)
71 \( 1 + (-2.32 - 1.68i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-13.1 + 6.71i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (12.3 - 8.96i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.04 - 6.60i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 - 11.1iT - 89T^{2} \)
97 \( 1 + (9.94 + 1.57i)T + (92.2 + 29.9i)T^{2} \)
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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

−10.99336385349699864173738986547, −9.871307966785591599389283532282, −8.952029715377732270503867964774, −7.88323314583486265651428579501, −7.01174732464521702817156902675, −6.33056325403507497998174202074, −4.55836993085716845666831982098, −3.91058712268314640137796224621, −2.57978886585380638635598520064, −0.826819909275441050266332469926, 2.10912057764032838454601264698, 3.13093608270837897943340971448, 4.55921972706955864231381302858, 5.91111888153509420981111538421, 6.68100562565841696127607956091, 7.27302287503843364324933157066, 8.429593395820813433236691135834, 9.433030561229320838009385325799, 10.41388283598074942483012504706, 11.29862319365506864016962903306

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