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Properties

Label 2-495-495.113-c1-0-25
Degree $2$
Conductor $495$
Sign $-0.0798 - 0.996i$
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  + (0.397 + 1.03i)2-s + (1.42 + 0.986i)3-s + (0.573 − 0.515i)4-s + (−2.17 − 0.534i)5-s + (−0.455 + 1.86i)6-s + (−0.976 + 1.50i)7-s + (2.73 + 1.39i)8-s + (1.05 + 2.80i)9-s + (−0.309 − 2.45i)10-s + (1.50 + 2.95i)11-s + (1.32 − 0.168i)12-s + (2.80 − 3.46i)13-s + (−1.94 − 0.413i)14-s + (−2.56 − 2.90i)15-s + (−0.194 + 1.85i)16-s + (0.726 + 4.58i)17-s + ⋯
L(s)  = 1  + (0.280 + 0.731i)2-s + (0.821 + 0.569i)3-s + (0.286 − 0.257i)4-s + (−0.970 − 0.239i)5-s + (−0.186 + 0.761i)6-s + (−0.369 + 0.568i)7-s + (0.967 + 0.493i)8-s + (0.350 + 0.936i)9-s + (−0.0977 − 0.777i)10-s + (0.453 + 0.891i)11-s + (0.382 − 0.0487i)12-s + (0.777 − 0.960i)13-s + (−0.519 − 0.110i)14-s + (−0.661 − 0.749i)15-s + (−0.0486 + 0.463i)16-s + (0.176 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0798 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0798 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.0798 - 0.996i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.0798 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43414 + 1.55365i\)
\(L(\frac12)\) \(\approx\) \(1.43414 + 1.55365i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.42 - 0.986i)T \)
5 \( 1 + (2.17 + 0.534i)T \)
11 \( 1 + (-1.50 - 2.95i)T \)
good2 \( 1 + (-0.397 - 1.03i)T + (-1.48 + 1.33i)T^{2} \)
7 \( 1 + (0.976 - 1.50i)T + (-2.84 - 6.39i)T^{2} \)
13 \( 1 + (-2.80 + 3.46i)T + (-2.70 - 12.7i)T^{2} \)
17 \( 1 + (-0.726 - 4.58i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (3.68 - 1.19i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.743 + 2.77i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.27 + 0.271i)T + (26.4 - 11.7i)T^{2} \)
31 \( 1 + (0.192 + 1.83i)T + (-30.3 + 6.44i)T^{2} \)
37 \( 1 + (4.56 + 8.95i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (0.238 - 1.12i)T + (-37.4 - 16.6i)T^{2} \)
43 \( 1 + (-2.90 + 10.8i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (4.17 - 0.218i)T + (46.7 - 4.91i)T^{2} \)
53 \( 1 + (-0.0628 + 0.396i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (2.49 + 2.76i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (0.116 - 1.11i)T + (-59.6 - 12.6i)T^{2} \)
67 \( 1 + (3.57 + 13.3i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-4.82 + 6.64i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-7.01 + 3.57i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (-1.34 + 3.02i)T + (-52.8 - 58.7i)T^{2} \)
83 \( 1 + (-2.63 - 3.25i)T + (-17.2 + 81.1i)T^{2} \)
89 \( 1 + 9.47T + 89T^{2} \)
97 \( 1 + (0.720 - 0.276i)T + (72.0 - 64.9i)T^{2} \)
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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

−10.83622037053574139099841366352, −10.45285198573128293445385979905, −9.170083801543682635320873475477, −8.310142603218156866054291920058, −7.70737889114013258986423563702, −6.60632145094918801711110880837, −5.56003184555125279945401154158, −4.43117525605925331940841990576, −3.57923353620934444672305518144, −2.02224774978069180649687167346, 1.22018697772613836652308309034, 2.83725338762838788295846074865, 3.58192946223307482617462463577, 4.33424505679507455103409586623, 6.57079301502097297353295386152, 7.00207285583724460799687686861, 8.026526230720110851949316612553, 8.784950994594145193066493818982, 9.920189541858257961627660189757, 11.09585687830206132937440521216

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