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Properties

Label 2-1800-360.347-c0-0-1
Degree $2$
Conductor $1800$
Sign $0.537 + 0.843i$
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 + (−0.965 + 0.258i)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 + (−0.965 − 0.258i)12-s + (0.500 + 0.866i)16-s + (0.707 − 0.707i)17-s + (−0.965 + 0.258i)18-s − 2i·19-s + (1.67 − 0.448i)22-s + (0.866 + 0.5i)24-s + (−0.707 + 0.707i)27-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (−0.965 + 0.258i)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 + (−0.965 − 0.258i)12-s + (0.500 + 0.866i)16-s + (0.707 − 0.707i)17-s + (−0.965 + 0.258i)18-s − 2i·19-s + (1.67 − 0.448i)22-s + (0.866 + 0.5i)24-s + (−0.707 + 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.537 + 0.843i$
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: \((2,\ 1800,\ (\ :0),\ 0.537 + 0.843i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4262008054\)
\(L(\frac12)\) \(\approx\) \(0.4262008054\)
\(L(1)\) 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 \( 1 + (0.965 - 0.258i)T \)
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 \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
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

−9.633223188353781841067614107963, −8.705421602390510980693184539276, −7.55047768372943128985081830439, −7.24140527858442053555557799772, −6.28804556928084107915404821410, −5.24239899129834796840710495390, −4.59791330116983235103219455424, −3.15949875164779551818262528937, −2.18548503439611525922434392025, −0.57918766092620623397402432434, 1.10618018633268612459486766940, 2.32397788424636289059520841682, 3.66223817578690572447361199050, 5.15962145941239097441565363768, 5.82829018472654942562648399346, 6.24145310025506188274193350150, 7.56375545638091301216340359317, 7.81242306443562643418285308384, 8.633693889713032370015961044124, 9.804862425546858840194659617230

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