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Properties

Label 2-495-45.32-c1-0-5
Degree $2$
Conductor $495$
Sign $-0.949 - 0.312i$
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  + (−1.44 − 0.386i)2-s + (1.29 + 1.14i)3-s + (0.198 + 0.114i)4-s + (−1.35 + 1.78i)5-s + (−1.43 − 2.15i)6-s + (−0.953 + 3.55i)7-s + (1.86 + 1.86i)8-s + (0.379 + 2.97i)9-s + (2.63 − 2.04i)10-s + (0.866 − 0.5i)11-s + (0.126 + 0.376i)12-s + (−1.27 − 4.74i)13-s + (2.75 − 4.76i)14-s + (−3.79 + 0.767i)15-s + (−2.20 − 3.81i)16-s + (−3.82 + 3.82i)17-s + ⋯
L(s)  = 1  + (−1.01 − 0.273i)2-s + (0.750 + 0.660i)3-s + (0.0992 + 0.0572i)4-s + (−0.604 + 0.796i)5-s + (−0.584 − 0.879i)6-s + (−0.360 + 1.34i)7-s + (0.660 + 0.660i)8-s + (0.126 + 0.991i)9-s + (0.834 − 0.646i)10-s + (0.261 − 0.150i)11-s + (0.0366 + 0.108i)12-s + (−0.352 − 1.31i)13-s + (0.735 − 1.27i)14-s + (−0.980 + 0.198i)15-s + (−0.550 − 0.953i)16-s + (−0.926 + 0.926i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.949 - 0.312i$
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: \((2,\ 495,\ (\ :1/2),\ -0.949 - 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0772630 + 0.481609i\)
\(L(\frac12)\) \(\approx\) \(0.0772630 + 0.481609i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.29 - 1.14i)T \)
5 \( 1 + (1.35 - 1.78i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good2 \( 1 + (1.44 + 0.386i)T + (1.73 + i)T^{2} \)
7 \( 1 + (0.953 - 3.55i)T + (-6.06 - 3.5i)T^{2} \)
13 \( 1 + (1.27 + 4.74i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (3.82 - 3.82i)T - 17iT^{2} \)
19 \( 1 + 6.45iT - 19T^{2} \)
23 \( 1 + (4.60 - 1.23i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.53 - 2.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.54 - 4.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.15 - 4.15i)T + 37iT^{2} \)
41 \( 1 + (5.98 + 3.45i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.85 - 1.03i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.88 + 0.505i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-9.36 - 9.36i)T + 53iT^{2} \)
59 \( 1 + (2.42 - 4.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.84 + 6.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.02 - 1.07i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 6.07iT - 71T^{2} \)
73 \( 1 + (-0.759 + 0.759i)T - 73iT^{2} \)
79 \( 1 + (4.14 - 2.39i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.710 + 2.65i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 8.84T + 89T^{2} \)
97 \( 1 + (1.59 - 5.94i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.89902005609889698629734349846, −10.40304464582473465199285958043, −9.479497983510894037034813350843, −8.698803851888063755647363733616, −8.228977113114248005176188205502, −7.15155961643866963036887903045, −5.72653618068701152165326269340, −4.53828732867385681736014185064, −3.12232732969510937726913589962, −2.30336127895366865086818504727, 0.37251321340577080679801789445, 1.74224362552151182606178257188, 3.90765224166514106600308256711, 4.29687945801181187900973948149, 6.48385375971284352380521612910, 7.29829103503447437222264037585, 7.81930549765575978610355520462, 8.705040269957232532489604266448, 9.467431956652991519572699110916, 10.06086823660412479245674592705

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