Properties

Label 2-1800-1.1-c3-0-13
Degree $2$
Conductor $1800$
Sign $1$
Analytic cond. $106.203$
Root an. cond. $10.3055$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 26·7-s + 59·11-s − 28·13-s + 5·17-s + 109·19-s − 194·23-s + 32·29-s + 10·31-s + 198·37-s − 117·41-s − 388·43-s − 68·47-s + 333·49-s − 18·53-s − 392·59-s − 710·61-s + 253·67-s + 612·71-s + 549·73-s − 1.53e3·77-s + 414·79-s − 121·83-s + 81·89-s + 728·91-s + 1.50e3·97-s + 234·101-s + 1.17e3·103-s + ⋯
L(s)  = 1  − 1.40·7-s + 1.61·11-s − 0.597·13-s + 0.0713·17-s + 1.31·19-s − 1.75·23-s + 0.204·29-s + 0.0579·31-s + 0.879·37-s − 0.445·41-s − 1.37·43-s − 0.211·47-s + 0.970·49-s − 0.0466·53-s − 0.864·59-s − 1.49·61-s + 0.461·67-s + 1.02·71-s + 0.880·73-s − 2.27·77-s + 0.589·79-s − 0.160·83-s + 0.0964·89-s + 0.838·91-s + 1.57·97-s + 0.230·101-s + 1.12·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(106.203\)
Root analytic conductor: \(10.3055\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.623182114\)
\(L(\frac12)\) \(\approx\) \(1.623182114\)
\(L(\frac{5}{2})\) 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 \( 1 + 26 T + p^{3} T^{2} \)
11 \( 1 - 59 T + p^{3} T^{2} \)
13 \( 1 + 28 T + p^{3} T^{2} \)
17 \( 1 - 5 T + p^{3} T^{2} \)
19 \( 1 - 109 T + p^{3} T^{2} \)
23 \( 1 + 194 T + p^{3} T^{2} \)
29 \( 1 - 32 T + p^{3} T^{2} \)
31 \( 1 - 10 T + p^{3} T^{2} \)
37 \( 1 - 198 T + p^{3} T^{2} \)
41 \( 1 + 117 T + p^{3} T^{2} \)
43 \( 1 + 388 T + p^{3} T^{2} \)
47 \( 1 + 68 T + p^{3} T^{2} \)
53 \( 1 + 18 T + p^{3} T^{2} \)
59 \( 1 + 392 T + p^{3} T^{2} \)
61 \( 1 + 710 T + p^{3} T^{2} \)
67 \( 1 - 253 T + p^{3} T^{2} \)
71 \( 1 - 612 T + p^{3} T^{2} \)
73 \( 1 - 549 T + p^{3} T^{2} \)
79 \( 1 - 414 T + p^{3} T^{2} \)
83 \( 1 + 121 T + p^{3} T^{2} \)
89 \( 1 - 81 T + p^{3} T^{2} \)
97 \( 1 - 1502 T + p^{3} T^{2} \)
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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

−9.170474303194529578552590647186, −8.111302615699579225715583640660, −7.21921641606883845754540527517, −6.44418372106866791886892183739, −5.96641922761082019807857810914, −4.75251869095824657444713976859, −3.73323162285852791753673394707, −3.14786917416481041115075918618, −1.84442289015236427275908077816, −0.59519584277076253382780006874, 0.59519584277076253382780006874, 1.84442289015236427275908077816, 3.14786917416481041115075918618, 3.73323162285852791753673394707, 4.75251869095824657444713976859, 5.96641922761082019807857810914, 6.44418372106866791886892183739, 7.21921641606883845754540527517, 8.111302615699579225715583640660, 9.170474303194529578552590647186

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