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Properties

Label 2-1800-1800.571-c0-0-0
Degree $2$
Conductor $1800$
Sign $-0.979 - 0.201i$
Analytic cond. $0.898317$
Root an. cond. $0.947795$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.207 + 0.978i)2-s + (0.809 + 0.587i)3-s + (−0.913 + 0.406i)4-s + (−0.406 − 0.913i)5-s + (−0.406 + 0.913i)6-s + (−0.866 + 0.5i)7-s + (−0.587 − 0.809i)8-s + (0.309 + 0.951i)9-s + (0.809 − 0.587i)10-s + (0.604 − 0.128i)11-s + (−0.978 − 0.207i)12-s + (−0.336 + 1.58i)13-s + (−0.669 − 0.743i)14-s + (0.207 − 0.978i)15-s + (0.669 − 0.743i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)2-s + (0.809 + 0.587i)3-s + (−0.913 + 0.406i)4-s + (−0.406 − 0.913i)5-s + (−0.406 + 0.913i)6-s + (−0.866 + 0.5i)7-s + (−0.587 − 0.809i)8-s + (0.309 + 0.951i)9-s + (0.809 − 0.587i)10-s + (0.604 − 0.128i)11-s + (−0.978 − 0.207i)12-s + (−0.336 + 1.58i)13-s + (−0.669 − 0.743i)14-s + (0.207 − 0.978i)15-s + (0.669 − 0.743i)16-s + (−0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.979 - 0.201i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ -0.979 - 0.201i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.047661309\)
\(L(\frac12)\) \(\approx\) \(1.047661309\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.207 - 0.978i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (0.406 + 0.913i)T \)
good7 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.604 + 0.128i)T + (0.913 - 0.406i)T^{2} \)
13 \( 1 + (0.336 - 1.58i)T + (-0.913 - 0.406i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.743 - 0.669i)T + (0.104 - 0.994i)T^{2} \)
29 \( 1 + (-0.614 - 0.0646i)T + (0.978 + 0.207i)T^{2} \)
31 \( 1 + (-0.994 + 0.104i)T + (0.978 - 0.207i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (-1.58 - 0.336i)T + (0.913 + 0.406i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.614 + 0.0646i)T + (0.978 + 0.207i)T^{2} \)
53 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.913 + 0.406i)T^{2} \)
61 \( 1 + (-0.207 - 0.978i)T + (-0.913 + 0.406i)T^{2} \)
67 \( 1 + (0.169 + 1.60i)T + (-0.978 + 0.207i)T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.614 + 0.0646i)T + (0.978 + 0.207i)T^{2} \)
83 \( 1 + (-0.564 - 0.251i)T + (0.669 + 0.743i)T^{2} \)
89 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.978 - 0.207i)T^{2} \)
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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

−9.503170520082510548853970251582, −8.810009667984859065319476163759, −8.524023452907433907597619345052, −7.57081672821671703803960974335, −6.59389886971813200037069872030, −5.93843669275691616154422410598, −4.70301509100674632861968531469, −4.20578577615222657163917852115, −3.51048013425426363447693155273, −2.02193224799647274025422948101, 0.65787479445534311836857715219, 2.45751851769454879973491021913, 2.86963483958751385997363460974, 3.78709737255910691061524252919, 4.53954607183980206653741083883, 6.13871269550395831840598914747, 6.67200840912887389296683816181, 7.62497057433924395507930582390, 8.377618318535007875934857708536, 9.213045177873701217670138299007

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