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Properties

Label 2-495-55.43-c1-0-7
Degree $2$
Conductor $495$
Sign $-0.953 - 0.301i$
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  + (1.91 + 1.91i)2-s + 5.35i·4-s + (−2.23 + 0.0440i)5-s + (2.40 + 2.40i)7-s + (−6.42 + 6.42i)8-s + (−4.37 − 4.20i)10-s + (−0.747 − 3.23i)11-s + (0.745 − 0.745i)13-s + 9.22i·14-s − 13.9·16-s + (0.546 + 0.546i)17-s − 0.734·19-s + (−0.235 − 11.9i)20-s + (4.76 − 7.62i)22-s + (3.12 + 3.12i)23-s + ⋯
L(s)  = 1  + (1.35 + 1.35i)2-s + 2.67i·4-s + (−0.999 + 0.0196i)5-s + (0.908 + 0.908i)7-s + (−2.27 + 2.27i)8-s + (−1.38 − 1.32i)10-s + (−0.225 − 0.974i)11-s + (0.206 − 0.206i)13-s + 2.46i·14-s − 3.48·16-s + (0.132 + 0.132i)17-s − 0.168·19-s + (−0.0526 − 2.67i)20-s + (1.01 − 1.62i)22-s + (0.652 + 0.652i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 - 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.953 - 0.301i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.953 - 0.301i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.378628 + 2.45071i\)
\(L(\frac12)\) \(\approx\) \(0.378628 + 2.45071i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 0.0440i)T \)
11 \( 1 + (0.747 + 3.23i)T \)
good2 \( 1 + (-1.91 - 1.91i)T + 2iT^{2} \)
7 \( 1 + (-2.40 - 2.40i)T + 7iT^{2} \)
13 \( 1 + (-0.745 + 0.745i)T - 13iT^{2} \)
17 \( 1 + (-0.546 - 0.546i)T + 17iT^{2} \)
19 \( 1 + 0.734T + 19T^{2} \)
23 \( 1 + (-3.12 - 3.12i)T + 23iT^{2} \)
29 \( 1 + 3.72T + 29T^{2} \)
31 \( 1 - 7.07T + 31T^{2} \)
37 \( 1 + (0.839 - 0.839i)T - 37iT^{2} \)
41 \( 1 - 4.44iT - 41T^{2} \)
43 \( 1 + (0.577 - 0.577i)T - 43iT^{2} \)
47 \( 1 + (-5.84 + 5.84i)T - 47iT^{2} \)
53 \( 1 + (-7.73 - 7.73i)T + 53iT^{2} \)
59 \( 1 + 7.74iT - 59T^{2} \)
61 \( 1 + 12.4iT - 61T^{2} \)
67 \( 1 + (4.58 - 4.58i)T - 67iT^{2} \)
71 \( 1 - 3.18T + 71T^{2} \)
73 \( 1 + (-2.23 + 2.23i)T - 73iT^{2} \)
79 \( 1 - 2.10T + 79T^{2} \)
83 \( 1 + (-4.87 + 4.87i)T - 83iT^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 + (-6.51 + 6.51i)T - 97iT^{2} \)
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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

−11.67587224711664679733078253224, −10.99632261036112392609229177623, −8.910411162766335543290743858652, −8.234463136131618032465672654912, −7.71472343902443081331011000646, −6.60798967371092797995262800344, −5.60221668615845849272399492448, −4.93736109797206951820315399488, −3.85372988323019463890693989781, −2.87616684939582642018460228677, 1.07813469491875214011062835556, 2.51495540827520515546340036172, 3.88317805312568060141329448382, 4.43436448203146221111269501483, 5.21611950481164485862196720340, 6.71990120868877103448405096559, 7.66747069446717541305893337702, 9.020826648090864808138396670464, 10.32953001008350627121992983887, 10.71838063788844955601453372050

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