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Properties

Label 2-1800-24.11-c1-0-66
Degree $2$
Conductor $1800$
Sign $0.577 + 0.816i$
Analytic cond. $14.3730$
Root an. cond. $3.79118$
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  + 1.41·2-s + 2.00·4-s + 1.16i·7-s + 2.82·8-s − 5.88i·11-s − 7.16i·13-s + 1.64i·14-s + 4.00·16-s − 6.32·19-s − 8.32i·22-s + 4.47·23-s − 10.1i·26-s + 2.32i·28-s + 5.65·32-s − 4.83i·37-s − 8.94·38-s + ⋯
L(s)  = 1  + 1.00·2-s + 1.00·4-s + 0.439i·7-s + 1.00·8-s − 1.77i·11-s − 1.98i·13-s + 0.439i·14-s + 1.00·16-s − 1.45·19-s − 1.77i·22-s + 0.932·23-s − 1.98i·26-s + 0.439i·28-s + 1.00·32-s − 0.795i·37-s − 1.45·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.291123644\)
\(L(\frac12)\) \(\approx\) \(3.291123644\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 1.16iT - 7T^{2} \)
11 \( 1 + 5.88iT - 11T^{2} \)
13 \( 1 + 7.16iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 6.32T + 19T^{2} \)
23 \( 1 - 4.47T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 4.83iT - 37T^{2} \)
41 \( 1 - 7.53iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 5.65T + 53T^{2} \)
59 \( 1 - 14.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 0.955iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−8.923652463939013006152894091807, −8.305143110446565088626780120473, −7.55587622005206366161343708612, −6.44250015075195157519806757395, −5.75932267399718115874435986300, −5.30784277659350137822849339354, −4.10128385940015863085871403759, −3.15311927583186665904352729241, −2.56886861950107551697750157215, −0.875248277867879607758384549861, 1.71091626222201092955486361486, 2.35872759527788046296008207703, 3.87091157302458082451684834766, 4.40609827893723820098980183541, 5.03777837945502452423828905875, 6.35165964833326471659288431559, 6.92832858541608514078678484106, 7.36866657775040103396179564628, 8.606794585282691973732667048351, 9.494692679060053665277525790072

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