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Properties

Label 2-495-55.43-c1-0-14
Degree $2$
Conductor $495$
Sign $0.974 - 0.223i$
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  + (0.857 + 0.857i)2-s − 0.528i·4-s + (−2.20 − 0.383i)5-s + (1.82 + 1.82i)7-s + (2.16 − 2.16i)8-s + (−1.56 − 2.21i)10-s + (2.70 + 1.92i)11-s + (4.37 − 4.37i)13-s + 3.13i·14-s + 2.66·16-s + (−2.10 − 2.10i)17-s − 1.00·19-s + (−0.202 + 1.16i)20-s + (0.666 + 3.96i)22-s + (3.04 + 3.04i)23-s + ⋯
L(s)  = 1  + (0.606 + 0.606i)2-s − 0.264i·4-s + (−0.985 − 0.171i)5-s + (0.690 + 0.690i)7-s + (0.766 − 0.766i)8-s + (−0.493 − 0.701i)10-s + (0.814 + 0.580i)11-s + (1.21 − 1.21i)13-s + 0.837i·14-s + 0.665·16-s + (−0.511 − 0.511i)17-s − 0.230·19-s + (−0.0453 + 0.260i)20-s + (0.142 + 0.845i)22-s + (0.634 + 0.634i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.974 - 0.223i$
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: \((2,\ 495,\ (\ :1/2),\ 0.974 - 0.223i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94853 + 0.220829i\)
\(L(\frac12)\) \(\approx\) \(1.94853 + 0.220829i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.20 + 0.383i)T \)
11 \( 1 + (-2.70 - 1.92i)T \)
good2 \( 1 + (-0.857 - 0.857i)T + 2iT^{2} \)
7 \( 1 + (-1.82 - 1.82i)T + 7iT^{2} \)
13 \( 1 + (-4.37 + 4.37i)T - 13iT^{2} \)
17 \( 1 + (2.10 + 2.10i)T + 17iT^{2} \)
19 \( 1 + 1.00T + 19T^{2} \)
23 \( 1 + (-3.04 - 3.04i)T + 23iT^{2} \)
29 \( 1 - 9.78T + 29T^{2} \)
31 \( 1 + 3.43T + 31T^{2} \)
37 \( 1 + (4.54 - 4.54i)T - 37iT^{2} \)
41 \( 1 + 4.36iT - 41T^{2} \)
43 \( 1 + (5.03 - 5.03i)T - 43iT^{2} \)
47 \( 1 + (2.08 - 2.08i)T - 47iT^{2} \)
53 \( 1 + (5.36 + 5.36i)T + 53iT^{2} \)
59 \( 1 - 4.42iT - 59T^{2} \)
61 \( 1 + 7.08iT - 61T^{2} \)
67 \( 1 + (10.3 - 10.3i)T - 67iT^{2} \)
71 \( 1 + 8.06T + 71T^{2} \)
73 \( 1 + (-2.48 + 2.48i)T - 73iT^{2} \)
79 \( 1 + 7.97T + 79T^{2} \)
83 \( 1 + (0.317 - 0.317i)T - 83iT^{2} \)
89 \( 1 - 0.471iT - 89T^{2} \)
97 \( 1 + (-1.60 + 1.60i)T - 97iT^{2} \)
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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

−11.13251846132129139976071582474, −10.20759426182031646179684939201, −8.925046202797863074966017724759, −8.226971294360781287678494392942, −7.19028929285153748840100193474, −6.31268290429793342572594279704, −5.20602433801259687539372145567, −4.51735057646413072380551431708, −3.33765153389003358650494475135, −1.29497145347362587027068020259, 1.50774948069254347157353083593, 3.23046502811389902998734570138, 4.12348228520984437249111961043, 4.60257954257019880901935345833, 6.39763119632847454137147351747, 7.24376288146256431366534899768, 8.422783083599831632332950029486, 8.755407804849522901825505412896, 10.65479645064093126483551038363, 11.05328403038939820935371518631

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