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Properties

Label 2-1800-120.59-c1-0-0
Degree $2$
Conductor $1800$
Sign $-0.858 - 0.513i$
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  + (1.35 − 0.408i)2-s + (1.66 − 1.10i)4-s − 3.40·7-s + (1.80 − 2.17i)8-s + 2.19i·11-s − 6.77·13-s + (−4.60 + 1.39i)14-s + (1.54 − 3.68i)16-s − 6.90·17-s − 1.49·19-s + (0.895 + 2.96i)22-s + 5.21i·23-s + (−9.16 + 2.76i)26-s + (−5.66 + 3.76i)28-s − 2.08·29-s + ⋯
L(s)  = 1  + (0.957 − 0.289i)2-s + (0.832 − 0.553i)4-s − 1.28·7-s + (0.637 − 0.770i)8-s + 0.660i·11-s − 1.87·13-s + (−1.23 + 0.371i)14-s + (0.387 − 0.921i)16-s − 1.67·17-s − 0.343·19-s + (0.190 + 0.632i)22-s + 1.08i·23-s + (−1.79 + 0.542i)26-s + (−1.07 + 0.711i)28-s − 0.387·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 - 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.858 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.858 - 0.513i$
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: \((2,\ 1800,\ (\ :1/2),\ -0.858 - 0.513i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.05379727276\)
\(L(\frac12)\) \(\approx\) \(0.05379727276\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 + 0.408i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 3.40T + 7T^{2} \)
11 \( 1 - 2.19iT - 11T^{2} \)
13 \( 1 + 6.77T + 13T^{2} \)
17 \( 1 + 6.90T + 17T^{2} \)
19 \( 1 + 1.49T + 19T^{2} \)
23 \( 1 - 5.21iT - 23T^{2} \)
29 \( 1 + 2.08T + 29T^{2} \)
31 \( 1 - 2.76iT - 31T^{2} \)
37 \( 1 - 5.06T + 37T^{2} \)
41 \( 1 + 1.10iT - 41T^{2} \)
43 \( 1 - 3.60iT - 43T^{2} \)
47 \( 1 + 11.0iT - 47T^{2} \)
53 \( 1 + 1.22iT - 53T^{2} \)
59 \( 1 + 11.5iT - 59T^{2} \)
61 \( 1 - 10.6iT - 61T^{2} \)
67 \( 1 + 10.8iT - 67T^{2} \)
71 \( 1 + 2.86T + 71T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 + 11.4iT - 79T^{2} \)
83 \( 1 - 8.71T + 83T^{2} \)
89 \( 1 - 1.40iT - 89T^{2} \)
97 \( 1 - 8.19iT - 97T^{2} \)
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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

−9.707152294735370012586159189023, −9.173827908145761714844552524187, −7.69839888024686941424283685257, −6.91459736032222763344898880400, −6.51880671397884719816836453463, −5.39248131392426585900206567376, −4.65587336732124513732008507528, −3.79710420602165461600581221749, −2.74278435522263862446010693773, −2.03275150063064318732734778760, 0.01230253821054051852653186593, 2.37413514071166216303222850295, 2.86624314614621170294852259062, 4.09702431990337310481023504074, 4.72216495743716625934347448792, 5.81493453351489888040858716638, 6.51147445499645302704026192029, 7.06173910039632656600281538407, 7.960495357118196262307672930642, 8.962883029714557685529133855540

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