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Properties

Label 2-1800-200.13-c0-0-1
Degree $2$
Conductor $1800$
Sign $0.562 - 0.827i$
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  + (0.156 + 0.987i)2-s + (−0.951 + 0.309i)4-s + (0.987 − 0.156i)5-s + (0.642 − 0.642i)7-s + (−0.453 − 0.891i)8-s + (0.309 + 0.951i)10-s + (−0.183 + 0.253i)11-s + (0.734 + 0.533i)14-s + (0.809 − 0.587i)16-s + (−0.891 + 0.453i)20-s + (−0.278 − 0.142i)22-s + (0.951 − 0.309i)25-s + (−0.412 + 0.809i)28-s + (0.437 + 1.34i)29-s + (0.587 − 1.80i)31-s + (0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)2-s + (−0.951 + 0.309i)4-s + (0.987 − 0.156i)5-s + (0.642 − 0.642i)7-s + (−0.453 − 0.891i)8-s + (0.309 + 0.951i)10-s + (−0.183 + 0.253i)11-s + (0.734 + 0.533i)14-s + (0.809 − 0.587i)16-s + (−0.891 + 0.453i)20-s + (−0.278 − 0.142i)22-s + (0.951 − 0.309i)25-s + (−0.412 + 0.809i)28-s + (0.437 + 1.34i)29-s + (0.587 − 1.80i)31-s + (0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.562 - 0.827i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ 0.562 - 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.398975244\)
\(L(\frac12)\) \(\approx\) \(1.398975244\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.156 - 0.987i)T \)
3 \( 1 \)
5 \( 1 + (-0.987 + 0.156i)T \)
good7 \( 1 + (-0.642 + 0.642i)T - iT^{2} \)
11 \( 1 + (0.183 - 0.253i)T + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.951 - 0.309i)T^{2} \)
17 \( 1 + (-0.587 + 0.809i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.951 + 0.309i)T^{2} \)
29 \( 1 + (-0.437 - 1.34i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.951 + 0.309i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.587 + 0.809i)T^{2} \)
53 \( 1 + (-1.04 - 0.533i)T + (0.587 + 0.809i)T^{2} \)
59 \( 1 + (1.59 - 1.16i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + (0.587 - 0.809i)T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.221 + 1.39i)T + (-0.951 + 0.309i)T^{2} \)
79 \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.280 + 0.550i)T + (-0.587 + 0.809i)T^{2} \)
89 \( 1 + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (1.58 + 0.809i)T + (0.587 + 0.809i)T^{2} \)
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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

−9.411440074794325409116555070630, −8.748953728879478679815935482235, −7.87059057616307715539446322796, −7.22759668471155498255293058426, −6.35808720330385550225398498982, −5.62727122096542171374623127872, −4.81015466566509993948954353957, −4.13302248647162462758695504557, −2.76675992347401116154458942270, −1.29783846086137134800509377705, 1.39882091854615404735618532194, 2.34942959515262295012425631345, 3.11773628661332102818381699068, 4.41334999543283377417505354108, 5.23792997639209218654857137137, 5.83033109346861447647125959615, 6.80829589090720425723162154559, 8.198522962643592444882301097360, 8.652901139858958242628552205750, 9.540463846991738299769735411311

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