Properties

Label 2-1800-8.5-c1-0-10
Degree $2$
Conductor $1800$
Sign $-0.306 - 0.951i$
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  + (−1.29 − 0.576i)2-s + (1.33 + 1.48i)4-s + 1.97·7-s + (−0.867 − 2.69i)8-s + 1.43i·11-s − 0.241i·13-s + (−2.55 − 1.13i)14-s + (−0.430 + 3.97i)16-s − 7.38·17-s + 3.04i·19-s + (0.824 − 1.84i)22-s + 0.874·23-s + (−0.139 + 0.311i)26-s + (2.64 + 2.94i)28-s + 9.07i·29-s + ⋯
L(s)  = 1  + (−0.913 − 0.407i)2-s + (0.667 + 0.744i)4-s + 0.747·7-s + (−0.306 − 0.951i)8-s + 0.431i·11-s − 0.0669i·13-s + (−0.682 − 0.304i)14-s + (−0.107 + 0.994i)16-s − 1.79·17-s + 0.697i·19-s + (0.175 − 0.393i)22-s + 0.182·23-s + (−0.0272 + 0.0611i)26-s + (0.499 + 0.556i)28-s + 1.68i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.306 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.306 - 0.951i$
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: \((2,\ 1800,\ (\ :1/2),\ -0.306 - 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5776460201\)
\(L(\frac12)\) \(\approx\) \(0.5776460201\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 + 0.576i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 1.97T + 7T^{2} \)
11 \( 1 - 1.43iT - 11T^{2} \)
13 \( 1 + 0.241iT - 13T^{2} \)
17 \( 1 + 7.38T + 17T^{2} \)
19 \( 1 - 3.04iT - 19T^{2} \)
23 \( 1 - 0.874T + 23T^{2} \)
29 \( 1 - 9.07iT - 29T^{2} \)
31 \( 1 + 7.44T + 31T^{2} \)
37 \( 1 - 8.81iT - 37T^{2} \)
41 \( 1 - 1.91T + 41T^{2} \)
43 \( 1 + 11.2iT - 43T^{2} \)
47 \( 1 + 3.34T + 47T^{2} \)
53 \( 1 + 9.20iT - 53T^{2} \)
59 \( 1 - 6.43iT - 59T^{2} \)
61 \( 1 - 4.57iT - 61T^{2} \)
67 \( 1 - 4.86iT - 67T^{2} \)
71 \( 1 - 8.21T + 71T^{2} \)
73 \( 1 - 4.12T + 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 - 12.3iT - 83T^{2} \)
89 \( 1 - 8.08T + 89T^{2} \)
97 \( 1 + 10.6T + 97T^{2} \)
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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

−9.443422019105097488652786898252, −8.677769433537877650072379605029, −8.225820941178381934293535052476, −7.13896863568168104920598622440, −6.75661406613353980431088948892, −5.45562516587524674874706891489, −4.45821108776312668399457771628, −3.48513000438817746203167674559, −2.26059746245436191671724008457, −1.46999755620462145334551293533, 0.28077978081483057188007821664, 1.76196450173472859397856046986, 2.64602463544845509111567834608, 4.21164284708796647128687880914, 5.07442722198735723964743801859, 6.05313912855157517836348627974, 6.73237211190042692745916107320, 7.64280113003288771967584465071, 8.218600582537886483142135343905, 9.118926933514869006459768719397

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