Properties

Label 2-42e2-63.47-c1-0-0
Degree $2$
Conductor $1764$
Sign $-0.958 + 0.285i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
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 + 0.956i)3-s − 2.86·5-s + (1.16 + 2.76i)9-s − 2.71i·11-s + (−3.18 + 1.84i)13-s + (−4.14 − 2.74i)15-s + (3.22 + 5.58i)17-s + (−2.73 − 1.58i)19-s − 2.99i·23-s + 3.22·25-s + (−0.956 + 5.10i)27-s + (−2.48 − 1.43i)29-s + (−8.26 − 4.77i)31-s + (2.59 − 3.91i)33-s + (−1.70 + 2.95i)37-s + ⋯
L(s)  = 1  + (0.833 + 0.552i)3-s − 1.28·5-s + (0.389 + 0.920i)9-s − 0.817i·11-s + (−0.884 + 0.510i)13-s + (−1.06 − 0.708i)15-s + (0.781 + 1.35i)17-s + (−0.628 − 0.362i)19-s − 0.623i·23-s + 0.645·25-s + (−0.184 + 0.982i)27-s + (−0.461 − 0.266i)29-s + (−1.48 − 0.857i)31-s + (0.451 − 0.681i)33-s + (−0.280 + 0.485i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.958 + 0.285i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.958 + 0.285i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2521133440\)
\(L(\frac12)\) \(\approx\) \(0.2521133440\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.44 - 0.956i)T \)
7 \( 1 \)
good5 \( 1 + 2.86T + 5T^{2} \)
11 \( 1 + 2.71iT - 11T^{2} \)
13 \( 1 + (3.18 - 1.84i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.22 - 5.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.73 + 1.58i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.99iT - 23T^{2} \)
29 \( 1 + (2.48 + 1.43i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (8.26 + 4.77i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.70 - 2.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.794 + 1.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.67 - 8.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.65 + 9.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.16 - 1.24i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.33 + 7.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.566 + 0.327i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.86 - 6.68i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.86iT - 71T^{2} \)
73 \( 1 + (11.0 - 6.39i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.59 + 4.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.92 - 13.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.14 + 5.45i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.2 + 7.62i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.684322007611688831511061452051, −8.737193446537061375852905056504, −8.219453790580932025918037838543, −7.63522891593455796739969622270, −6.74676960555441127515080959430, −5.54754351552828657378045415515, −4.49226704805836131140718547815, −3.85368591092834969300332037581, −3.15426890398007845009044619307, −1.91210681549878312219361440179, 0.080364077759506216620358376719, 1.67903624095749685919384392502, 2.91287112503336801966309943467, 3.62536106615101353616978340648, 4.57651047219939066384106984431, 5.53729062596056561967151089911, 7.01631554931933008169569503783, 7.35258727736146959850215200592, 7.87198003614131412252819416039, 8.771700018078527996261627629552

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