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Properties

Label 2-1800-8.5-c1-0-60
Degree $2$
Conductor $1800$
Sign $1$
Analytic cond. $14.3730$
Root an. cond. $3.79118$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + 2.00·4-s + 2.82·8-s − 4.47i·11-s + 6.32i·13-s + 4.00·16-s + 2.82·17-s − 6.32i·22-s + 5.65·23-s + 8.94i·26-s + 4.47i·29-s + 2·31-s + 5.65·32-s + 4.00·34-s − 6.32i·37-s + ⋯
L(s)  = 1  + 1.00·2-s + 1.00·4-s + 1.00·8-s − 1.34i·11-s + 1.75i·13-s + 1.00·16-s + 0.685·17-s − 1.34i·22-s + 1.17·23-s + 1.75i·26-s + 0.830i·29-s + 0.359·31-s + 1.00·32-s + 0.685·34-s − 1.03i·37-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.642136967\)
\(L(\frac12)\) \(\approx\) \(3.642136967\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + 4.47iT - 11T^{2} \)
13 \( 1 - 6.32iT - 13T^{2} \)
17 \( 1 - 2.82T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 5.65T + 23T^{2} \)
29 \( 1 - 4.47iT - 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 6.32iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 12.6iT - 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 4.47iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 12.6iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 14T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.117966693328301253255379227158, −8.569434023416496913515125650197, −7.35257843648195950322137341860, −6.84400101793898608959813966454, −5.90842224045289553594375133979, −5.26724058838719926595848995352, −4.22620944874835906214169105691, −3.49739731886081680667106932138, −2.50943519037463279362645708739, −1.25508048454219950982752110544, 1.24025684309267722243947601469, 2.60570161908968463137462635355, 3.29376860517439215172738799966, 4.47753294219461311597704097934, 5.09785867125132723204938054807, 5.92232556484349220390557285734, 6.79136744912557205867992485068, 7.66096688729113492808640862789, 8.100130925687431853901520614265, 9.527032537369586279145738411508

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