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Properties

Label 2-1800-8.5-c1-0-28
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  + (0.366 + 1.36i)2-s + (−1.73 + i)4-s + 2.73·7-s + (−2 − 1.99i)8-s + 2i·11-s − 3.46i·13-s + (1 + 3.73i)14-s + (1.99 − 3.46i)16-s − 3.46·17-s + 7.46i·19-s + (−2.73 + 0.732i)22-s + 4.19·23-s + (4.73 − 1.26i)26-s + (−4.73 + 2.73i)28-s + 6.92i·29-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.5i)4-s + 1.03·7-s + (−0.707 − 0.707i)8-s + 0.603i·11-s − 0.960i·13-s + (0.267 + 0.997i)14-s + (0.499 − 0.866i)16-s − 0.840·17-s + 1.71i·19-s + (−0.582 + 0.156i)22-s + 0.874·23-s + (0.928 − 0.248i)26-s + (−0.894 + 0.516i)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

\(L(1)\) \(\approx\) \(1.759314178\)
\(L(\frac12)\) \(\approx\) \(1.759314178\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 2.73T + 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 - 7.46iT - 19T^{2} \)
23 \( 1 - 4.19T + 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 - 1.46T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 5.46T + 41T^{2} \)
43 \( 1 - 8.73iT - 43T^{2} \)
47 \( 1 - 6.73T + 47T^{2} \)
53 \( 1 - 4.53iT - 53T^{2} \)
59 \( 1 - 0.535iT - 59T^{2} \)
61 \( 1 - 4.92iT - 61T^{2} \)
67 \( 1 + 7.26iT - 67T^{2} \)
71 \( 1 - 1.46T + 71T^{2} \)
73 \( 1 + 0.535T + 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 - 4.73iT - 83T^{2} \)
89 \( 1 - 4.92T + 89T^{2} \)
97 \( 1 + 6.39T + 97T^{2} \)
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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

−9.329178622575976287238617960733, −8.545290947626753631773761318318, −7.88319925034561733633239607497, −7.32895478578751269488575577706, −6.35342789167998192351429089688, −5.50413984072621867437478269186, −4.81287414242273894467219978595, −4.03650263376282594148551657355, −2.87826714235681184658344459995, −1.34199150637179295573846830704, 0.65892818651162965730428811272, 1.97631472798150239608124480479, 2.76062461967093825512111621067, 4.08047426656114933265909162256, 4.64226078198880089883451875933, 5.45422510116094619496675468821, 6.50110993518538510628342355347, 7.44961894939055142553507036800, 8.666814812371835171078460023961, 8.873971492019650506102202208267

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