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Properties

Label 2-1800-120.59-c1-0-23
Degree $2$
Conductor $1800$
Sign $0.0998 - 0.995i$
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  + (−0.933 + 1.06i)2-s + (−0.257 − 1.98i)4-s − 1.41·7-s + (2.34 + 1.57i)8-s + 2.31i·11-s + 5.14·13-s + (1.32 − 1.50i)14-s + (−3.86 + 1.02i)16-s + 5.10·17-s − 8.24·19-s + (−2.46 − 2.16i)22-s + 0.969i·23-s + (−4.80 + 5.46i)26-s + (0.364 + 2.80i)28-s + 3.28·29-s + ⋯
L(s)  = 1  + (−0.660 + 0.751i)2-s + (−0.128 − 0.991i)4-s − 0.534·7-s + (0.830 + 0.557i)8-s + 0.699i·11-s + 1.42·13-s + (0.352 − 0.401i)14-s + (−0.966 + 0.255i)16-s + 1.23·17-s − 1.89·19-s + (−0.525 − 0.461i)22-s + 0.202i·23-s + (−0.942 + 1.07i)26-s + (0.0688 + 0.530i)28-s + 0.609·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0998 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0998 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.0998 - 0.995i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ 0.0998 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.086306121\)
\(L(\frac12)\) \(\approx\) \(1.086306121\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.933 - 1.06i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 1.41T + 7T^{2} \)
11 \( 1 - 2.31iT - 11T^{2} \)
13 \( 1 - 5.14T + 13T^{2} \)
17 \( 1 - 5.10T + 17T^{2} \)
19 \( 1 + 8.24T + 19T^{2} \)
23 \( 1 - 0.969iT - 23T^{2} \)
29 \( 1 - 3.28T + 29T^{2} \)
31 \( 1 + 5.10iT - 31T^{2} \)
37 \( 1 - 3.69T + 37T^{2} \)
41 \( 1 + 4.59iT - 41T^{2} \)
43 \( 1 + 3.21iT - 43T^{2} \)
47 \( 1 - 9.52iT - 47T^{2} \)
53 \( 1 - 7.21iT - 53T^{2} \)
59 \( 1 + 0.862iT - 59T^{2} \)
61 \( 1 - 4.63iT - 61T^{2} \)
67 \( 1 - 5.28iT - 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 - 12.5iT - 73T^{2} \)
79 \( 1 - 8.01iT - 79T^{2} \)
83 \( 1 - 7.38T + 83T^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 + 15.7iT - 97T^{2} \)
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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

−9.379267040563376013420420465015, −8.576978075981726403574525506747, −7.976833432499984156865710135932, −7.09988282606624761333679156431, −6.24350354268189985840688155158, −5.83668342686500081730352966405, −4.61753704866235060143125431364, −3.74601786395436671482480603891, −2.27426737788142189255060414547, −1.01492703458980979242460157542, 0.64462743479282980507737711409, 1.86305266527008077155983619334, 3.17911057025679005686298636578, 3.66423831380089198113336053706, 4.79503706775769245940010055895, 6.16065584432124455150423036927, 6.61445955574416002205824379388, 7.914474349238323140164060093988, 8.429022802315309682465513216959, 9.033688376065884901817574009122

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