Properties

Label 2-1800-360.227-c0-0-5
Degree $2$
Conductor $1800$
Sign $0.514 + 0.857i$
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  + (0.258 + 0.965i)2-s + (0.258 − 0.965i)3-s + (−0.866 + 0.499i)4-s + 6-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−1.5 − 0.866i)11-s + (0.258 + 0.965i)12-s + (0.500 − 0.866i)16-s + (0.707 − 0.707i)17-s + (0.258 − 0.965i)18-s − 2i·19-s + (0.448 − 1.67i)22-s + (−0.866 + 0.5i)24-s + (−0.707 + 0.707i)27-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (0.258 − 0.965i)3-s + (−0.866 + 0.499i)4-s + 6-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−1.5 − 0.866i)11-s + (0.258 + 0.965i)12-s + (0.500 − 0.866i)16-s + (0.707 − 0.707i)17-s + (0.258 − 0.965i)18-s − 2i·19-s + (0.448 − 1.67i)22-s + (−0.866 + 0.5i)24-s + (−0.707 + 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.514 + 0.857i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ 0.514 + 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9333247558\)
\(L(\frac12)\) \(\approx\) \(0.9333247558\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 \)
good7 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
19 \( 1 + 2iT - T^{2} \)
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 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
89 \( 1 + 1.73T + T^{2} \)
97 \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \)
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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

−8.987524896066500925783377855926, −8.324329209469722851273016462814, −7.63678263843518123324689056677, −7.09531276876579411350495386025, −6.20204284715359451763261878440, −5.44090625738376086517404025227, −4.73158218509166383123112270314, −3.21463747184106147804899495600, −2.65471491415426207051933902326, −0.61138714427806882444659921441, 1.84309416087299561997081507059, 2.81193297409781177621192577115, 3.74276156036104198144548799378, 4.42899211420397330370308116252, 5.44627272255301423756405452260, 5.81591339737399440000860977868, 7.58493557241131463451346516047, 8.199909121451316829877861190767, 9.024489606336043090030199032315, 9.968938736573414551638737167379

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