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Properties

Label 2-1800-5.2-c2-0-41
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  + (−6.70 − 6.70i)7-s + 1.40·11-s + (14.4 − 14.4i)13-s + (−2.40 − 2.40i)17-s − 22.8i·19-s + (−0.104 + 0.104i)23-s + 45.6i·29-s − 2.59·31-s + (−10.6 − 10.6i)37-s + 44.6·41-s + (−26.7 + 26.7i)43-s + (−10.4 − 10.4i)47-s + 40.8i·49-s + (−3 + 3i)53-s − 41.1i·59-s + ⋯
L(s)  = 1  + (−0.957 − 0.957i)7-s + 0.127·11-s + (1.10 − 1.10i)13-s + (−0.141 − 0.141i)17-s − 1.20i·19-s + (−0.00455 + 0.00455i)23-s + 1.57i·29-s − 0.0837·31-s + (−0.286 − 0.286i)37-s + 1.08·41-s + (−0.620 + 0.620i)43-s + (−0.223 − 0.223i)47-s + 0.833i·49-s + (−0.0566 + 0.0566i)53-s − 0.698i·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

\(L(\frac{3}{2})\) \(\approx\) \(0.8299730623\)
\(L(\frac12)\) \(\approx\) \(0.8299730623\)
\(L(2)\) 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 \( 1 + (6.70 + 6.70i)T + 49iT^{2} \)
11 \( 1 - 1.40T + 121T^{2} \)
13 \( 1 + (-14.4 + 14.4i)T - 169iT^{2} \)
17 \( 1 + (2.40 + 2.40i)T + 289iT^{2} \)
19 \( 1 + 22.8iT - 361T^{2} \)
23 \( 1 + (0.104 - 0.104i)T - 529iT^{2} \)
29 \( 1 - 45.6iT - 841T^{2} \)
31 \( 1 + 2.59T + 961T^{2} \)
37 \( 1 + (10.6 + 10.6i)T + 1.36e3iT^{2} \)
41 \( 1 - 44.6T + 1.68e3T^{2} \)
43 \( 1 + (26.7 - 26.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (10.4 + 10.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (3 - 3i)T - 2.80e3iT^{2} \)
59 \( 1 + 41.1iT - 3.48e3T^{2} \)
61 \( 1 + 57.4T + 3.72e3T^{2} \)
67 \( 1 + (34.7 + 34.7i)T + 4.48e3iT^{2} \)
71 \( 1 + 45.4T + 5.04e3T^{2} \)
73 \( 1 + (11.3 - 11.3i)T - 5.32e3iT^{2} \)
79 \( 1 + 86.4iT - 6.24e3T^{2} \)
83 \( 1 + (81.7 - 81.7i)T - 6.88e3iT^{2} \)
89 \( 1 + 91.2iT - 7.92e3T^{2} \)
97 \( 1 + (49 + 49i)T + 9.40e3iT^{2} \)
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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

−8.833784116802552420237811992131, −7.88373274972811457530267020842, −7.05260645865586335592232541027, −6.45506161908071789977027845316, −5.54355437040808102437114130683, −4.52402621526787940427854064511, −3.51647209779121165188605077679, −2.93191864009995726050940548497, −1.26649057513744039559556033148, −0.22800229052870168057978985555, 1.49118433909698644122611785031, 2.56130948527660881748377157386, 3.62686828461229354140582620667, 4.35687413020530369802996698225, 5.78010719643977128017061771101, 6.09932393733534399902124490191, 6.92130433041378102948440244641, 8.033005108663942761560320911747, 8.760748618838620685762109844307, 9.392454178984170028663060206491

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