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Properties

Label 2-1800-8.5-c1-0-72
Degree $2$
Conductor $1800$
Sign $-0.707 + 0.707i$
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  + (−1.36 + 0.366i)2-s + (1.73 − i)4-s − 0.732·7-s + (−1.99 + 2i)8-s − 2i·11-s − 3.46i·13-s + (1 − 0.267i)14-s + (1.99 − 3.46i)16-s + 3.46·17-s − 0.535i·19-s + (0.732 + 2.73i)22-s − 6.19·23-s + (1.26 + 4.73i)26-s + (−1.26 + 0.732i)28-s + 6.92i·29-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.5i)4-s − 0.276·7-s + (−0.707 + 0.707i)8-s − 0.603i·11-s − 0.960i·13-s + (0.267 − 0.0716i)14-s + (0.499 − 0.866i)16-s + 0.840·17-s − 0.122i·19-s + (0.156 + 0.582i)22-s − 1.29·23-s + (0.248 + 0.928i)26-s + (−0.239 + 0.138i)28-s + 1.28i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.707 + 0.707i$
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: \((2,\ 1800,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4088336811\)
\(L(\frac12)\) \(\approx\) \(0.4088336811\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 0.732T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + 0.535iT - 19T^{2} \)
23 \( 1 + 6.19T + 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 + 5.46T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 1.46T + 41T^{2} \)
43 \( 1 + 5.26iT - 43T^{2} \)
47 \( 1 - 3.26T + 47T^{2} \)
53 \( 1 + 11.4iT - 53T^{2} \)
59 \( 1 + 7.46iT - 59T^{2} \)
61 \( 1 - 8.92iT - 61T^{2} \)
67 \( 1 - 10.7iT - 67T^{2} \)
71 \( 1 + 5.46T + 71T^{2} \)
73 \( 1 + 7.46T + 73T^{2} \)
79 \( 1 + 1.07T + 79T^{2} \)
83 \( 1 + 1.26iT - 83T^{2} \)
89 \( 1 + 8.92T + 89T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−8.870598007767415978440369909838, −8.258618941157256861134978146173, −7.53225238808586563492069854469, −6.76402669666804394415958236972, −5.77194721436323308792830874430, −5.33431407809610082571949379537, −3.69654866653822126223870902057, −2.84025344071984475779543777462, −1.54681971372107638618552544593, −0.20843093892489246251001890974, 1.47112066662237462857561876752, 2.40869432854448879509553710864, 3.56501421835816472023405821624, 4.45019683532992169697799807571, 5.85745124893987466333253314909, 6.49950761551401451318023946466, 7.47311293747746769888696871426, 7.939960560995314752770575416385, 8.947625053477352915483551793877, 9.623883639691834196312563946893

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