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Properties

Label 2-1800-5.2-c2-0-37
Degree $2$
Conductor $1800$
Sign $-0.973 + 0.229i$
Analytic cond. $49.0464$
Root an. cond. $7.00331$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.550 + 0.550i)7-s + 1.55·11-s + (−9.55 + 9.55i)13-s + (11.1 + 11.1i)17-s − 12.6i·19-s + (−21.3 + 21.3i)23-s − 44.0i·29-s − 44.4·31-s + (−20.6 − 20.6i)37-s + 48.2·41-s + (36.2 − 36.2i)43-s + (42.5 + 42.5i)47-s − 48.3i·49-s + (−54.4 + 54.4i)53-s + 47.4i·59-s + ⋯
L(s)  = 1  + (0.0786 + 0.0786i)7-s + 0.140·11-s + (−0.734 + 0.734i)13-s + (0.655 + 0.655i)17-s − 0.668i·19-s + (−0.928 + 0.928i)23-s − 1.51i·29-s − 1.43·31-s + (−0.558 − 0.558i)37-s + 1.17·41-s + (0.844 − 0.844i)43-s + (0.905 + 0.905i)47-s − 0.987i·49-s + (−1.02 + 1.02i)53-s + 0.804i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.973 + 0.229i$
Analytic conductor: \(49.0464\)
Root analytic conductor: \(7.00331\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1),\ -0.973 + 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.05145350464\)
\(L(\frac12)\) \(\approx\) \(0.05145350464\)
\(L(2)\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.550 - 0.550i)T + 49iT^{2} \)
11 \( 1 - 1.55T + 121T^{2} \)
13 \( 1 + (9.55 - 9.55i)T - 169iT^{2} \)
17 \( 1 + (-11.1 - 11.1i)T + 289iT^{2} \)
19 \( 1 + 12.6iT - 361T^{2} \)
23 \( 1 + (21.3 - 21.3i)T - 529iT^{2} \)
29 \( 1 + 44.0iT - 841T^{2} \)
31 \( 1 + 44.4T + 961T^{2} \)
37 \( 1 + (20.6 + 20.6i)T + 1.36e3iT^{2} \)
41 \( 1 - 48.2T + 1.68e3T^{2} \)
43 \( 1 + (-36.2 + 36.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-42.5 - 42.5i)T + 2.20e3iT^{2} \)
53 \( 1 + (54.4 - 54.4i)T - 2.80e3iT^{2} \)
59 \( 1 - 47.4iT - 3.48e3T^{2} \)
61 \( 1 + 59.8T + 3.72e3T^{2} \)
67 \( 1 + (81.2 + 81.2i)T + 4.48e3iT^{2} \)
71 \( 1 + 87.5T + 5.04e3T^{2} \)
73 \( 1 + (75.9 - 75.9i)T - 5.32e3iT^{2} \)
79 \( 1 + 97.3iT - 6.24e3T^{2} \)
83 \( 1 + (-41.0 + 41.0i)T - 6.88e3iT^{2} \)
89 \( 1 - 52.2iT - 7.92e3T^{2} \)
97 \( 1 + (-37 - 37i)T + 9.40e3iT^{2} \)
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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

−8.974114401807498415057137623942, −7.61063129243483088386478119457, −7.47849419773222499862510265798, −6.17214805102423615065004134341, −5.62297563679052352177567679685, −4.48000389470065501016256752998, −3.79087105233315119668194643659, −2.54456828132656011166500781762, −1.58278448050718673906226123206, −0.01308342387715490128779479009, 1.37093775312197199164667815682, 2.62344616557468518279611848141, 3.53095043793525641951449354192, 4.58930637426654127748575375620, 5.42454499159831172182739244489, 6.18048795735791672903430111650, 7.29912071740533588986665751182, 7.72752968134667231208659487678, 8.700456907663335554506601808656, 9.455518702824833842454817503805

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