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Properties

Label 2-1800-360.347-c0-0-0
Degree $2$
Conductor $1800$
Sign $0.157 - 0.987i$
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  + (−0.965 − 0.258i)2-s + (0.707 + 0.707i)3-s + (0.866 + 0.499i)4-s + (−0.500 − 0.866i)6-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (0.258 + 0.965i)12-s + (0.500 + 0.866i)16-s + (−1.41 + 1.41i)17-s + (0.258 − 0.965i)18-s + i·19-s − i·24-s + (−0.707 + 0.707i)27-s + (−0.258 − 0.965i)32-s + (1.73 − 1.00i)34-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.707 + 0.707i)3-s + (0.866 + 0.499i)4-s + (−0.500 − 0.866i)6-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (0.258 + 0.965i)12-s + (0.500 + 0.866i)16-s + (−1.41 + 1.41i)17-s + (0.258 − 0.965i)18-s + i·19-s − i·24-s + (−0.707 + 0.707i)27-s + (−0.258 − 0.965i)32-s + (1.73 − 1.00i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.157 - 0.987i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ 0.157 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8540677494\)
\(L(\frac12)\) \(\approx\) \(0.8540677494\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
89 \( 1 + 1.73T + T^{2} \)
97 \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \)
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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

−9.638324371842780737145427014514, −8.855471360968843814644748775508, −8.271047501821335065893000953774, −7.67767885782264449158134199280, −6.60967339620055346936900928909, −5.77761961606681436619796616964, −4.36827592010468874051292735820, −3.70517560271986664768532305555, −2.60200613973459279840209623612, −1.72143979289664973694696835527, 0.798092424088806672610416449031, 2.27168564724223764924878122374, 2.78477946707718283872388883643, 4.27345797226908851696189466642, 5.50537825044522322862822721411, 6.51480026656485960283298917010, 7.12314190617666244428848427496, 7.64237066154831658282875224320, 8.659423147857598911148490805641, 9.120973775179019937332265462652

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