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Properties

Label 2-1800-8.3-c0-0-0
Degree $2$
Conductor $1800$
Sign $-i$
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  + i·2-s − 4-s − i·8-s + 16-s + 2·19-s + 2i·23-s + i·32-s + 2i·38-s − 2·46-s + 2i·47-s + 49-s − 2i·53-s − 64-s − 2·76-s − 2i·92-s + ⋯
L(s)  = 1  + i·2-s − 4-s − i·8-s + 16-s + 2·19-s + 2i·23-s + i·32-s + 2i·38-s − 2·46-s + 2i·47-s + 49-s − 2i·53-s − 64-s − 2·76-s − 2i·92-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.041426874\)
\(L(\frac12)\) \(\approx\) \(1.041426874\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.538598294078989934211695230641, −8.870074630696383589361384768927, −7.74566233892626512165522075739, −7.49769466758534946154793458173, −6.52032910451760539452825884094, −5.56430304165349755863178233185, −5.11051996851060141493130748150, −3.92464699660655954136999064679, −3.11307463694740835129736200018, −1.29112818170464682582566971667, 0.973760871011093482261064088893, 2.33506781760078880709984307303, 3.19218900977646907267409218565, 4.16635711415261762961427530771, 5.03289447759258580351166390394, 5.81784610300805052843881713215, 6.98907380353146703420954012951, 7.87829171227281146084586361321, 8.717447970423711972151597332696, 9.338893165696946189430468179389

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