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Properties

Label 2-495-45.32-c1-0-54
Degree $2$
Conductor $495$
Sign $0.959 + 0.281i$
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  + (2.71 + 0.726i)2-s + (−0.190 − 1.72i)3-s + (5.09 + 2.94i)4-s + (−1.69 − 1.45i)5-s + (0.735 − 4.80i)6-s + (0.550 − 2.05i)7-s + (7.70 + 7.70i)8-s + (−2.92 + 0.654i)9-s + (−3.54 − 5.18i)10-s + (0.866 − 0.5i)11-s + (4.09 − 9.33i)12-s + (−0.143 − 0.536i)13-s + (2.98 − 5.16i)14-s + (−2.18 + 3.19i)15-s + (9.42 + 16.3i)16-s + (−0.0807 + 0.0807i)17-s + ⋯
L(s)  = 1  + (1.91 + 0.513i)2-s + (−0.109 − 0.993i)3-s + (2.54 + 1.47i)4-s + (−0.758 − 0.651i)5-s + (0.300 − 1.96i)6-s + (0.207 − 0.775i)7-s + (2.72 + 2.72i)8-s + (−0.975 + 0.218i)9-s + (−1.12 − 1.63i)10-s + (0.261 − 0.150i)11-s + (1.18 − 2.69i)12-s + (−0.0398 − 0.148i)13-s + (0.797 − 1.38i)14-s + (−0.564 + 0.825i)15-s + (2.35 + 4.07i)16-s + (−0.0195 + 0.0195i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.959 + 0.281i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.959 + 0.281i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.79152 - 0.544250i\)
\(L(\frac12)\) \(\approx\) \(3.79152 - 0.544250i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.190 + 1.72i)T \)
5 \( 1 + (1.69 + 1.45i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good2 \( 1 + (-2.71 - 0.726i)T + (1.73 + i)T^{2} \)
7 \( 1 + (-0.550 + 2.05i)T + (-6.06 - 3.5i)T^{2} \)
13 \( 1 + (0.143 + 0.536i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.0807 - 0.0807i)T - 17iT^{2} \)
19 \( 1 - 3.56iT - 19T^{2} \)
23 \( 1 + (6.28 - 1.68i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.77 - 4.80i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.67 + 2.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.38 + 5.38i)T + 37iT^{2} \)
41 \( 1 + (4.76 + 2.74i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.50 - 0.939i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (0.647 + 0.173i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (8.50 + 8.50i)T + 53iT^{2} \)
59 \( 1 + (3.70 - 6.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.32 - 2.29i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.03 - 0.814i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.78iT - 71T^{2} \)
73 \( 1 + (-9.56 + 9.56i)T - 73iT^{2} \)
79 \( 1 + (0.571 - 0.330i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.21 + 8.24i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 8.91T + 89T^{2} \)
97 \( 1 + (1.14 - 4.27i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.46798575635814708790068921891, −10.61784146442921236102131637732, −8.440428428549242946345534550442, −7.74309816510243196803195545986, −7.12698307258666033826217180297, −6.13049151143883893847204022623, −5.26062316699467255866951734222, −4.19650501179449738634723162554, −3.35464750829301903190994526276, −1.72461535227242215445109936803, 2.39610702479143286770992988676, 3.29463159953469487689246006080, 4.26079803959330285997394968662, 4.93989592528305809046840299719, 6.03848867889887166324293443594, 6.77082363563047898976499142142, 8.168622425521889028205949858006, 9.703939679776357426762358478737, 10.54120651353677597404937729494, 11.26039277187870076442594075441

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