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Properties

Label 2-1800-40.37-c0-0-1
Degree $2$
Conductor $1800$
Sign $-0.229 - 0.973i$
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.707i)8-s − 1.00·16-s + (1.41 + 1.41i)17-s + (−1.41 + 1.41i)23-s + 2·31-s + (−0.707 − 0.707i)32-s + 2.00i·34-s − 2.00·46-s + (−1.41 − 1.41i)47-s − i·49-s + (1.41 + 1.41i)62-s − 1.00i·64-s + (−1.41 + 1.41i)68-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.707i)8-s − 1.00·16-s + (1.41 + 1.41i)17-s + (−1.41 + 1.41i)23-s + 2·31-s + (−0.707 − 0.707i)32-s + 2.00i·34-s − 2.00·46-s + (−1.41 − 1.41i)47-s − i·49-s + (1.41 + 1.41i)62-s − 1.00i·64-s + (−1.41 + 1.41i)68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ -0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.591308679\)
\(L(\frac12)\) \(\approx\) \(1.591308679\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.770417280706322712897747219875, −8.491808507184014094783053985041, −8.082248715938173606792201093989, −7.32160137116835459885549301045, −6.28900125648010999533508987003, −5.79097085703071924061292360397, −4.89531623028534617304205670325, −3.86640789899309834830782873351, −3.24639755383553595808473518136, −1.82851827031531021328183803814, 1.02645937944997057442079104242, 2.46030639874335666833929895597, 3.17409481205816572501711134367, 4.31945879374057821403942592209, 4.95342661432611447254647841279, 5.94106425167865512117961146499, 6.58374691860214978701536237433, 7.67975862457911662768031247318, 8.496517330671342701795695318947, 9.714668638991628785525666823295

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