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Properties

Label 2-1800-5.3-c2-0-44
Degree $2$
Conductor $1800$
Sign $-0.973 - 0.229i$
Analytic cond. $49.0464$
Root an. cond. $7.00331$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.44 − 5.44i)7-s + 6.44·11-s + (−14.4 − 14.4i)13-s + (−23.1 + 23.1i)17-s − 16.6i·19-s + (−6.65 − 6.65i)23-s − 0.0454i·29-s + 4.49·31-s + (−35.3 + 35.3i)37-s − 20.2·41-s + (−32.2 − 32.2i)43-s + (−50.5 + 50.5i)47-s − 10.3i·49-s + (−5.50 − 5.50i)53-s + 55.4i·59-s + ⋯
L(s)  = 1  + (0.778 − 0.778i)7-s + 0.586·11-s + (−1.11 − 1.11i)13-s + (−1.36 + 1.36i)17-s − 0.878i·19-s + (−0.289 − 0.289i)23-s − 0.00156i·29-s + 0.144·31-s + (−0.955 + 0.955i)37-s − 0.494·41-s + (−0.750 − 0.750i)43-s + (−1.07 + 1.07i)47-s − 0.212i·49-s + (−0.103 − 0.103i)53-s + 0.939i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.973 - 0.229i$
Analytic conductor: \(49.0464\)
Root analytic conductor: \(7.00331\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1),\ -0.973 - 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2034426751\)
\(L(\frac12)\) \(\approx\) \(0.2034426751\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-5.44 + 5.44i)T - 49iT^{2} \)
11 \( 1 - 6.44T + 121T^{2} \)
13 \( 1 + (14.4 + 14.4i)T + 169iT^{2} \)
17 \( 1 + (23.1 - 23.1i)T - 289iT^{2} \)
19 \( 1 + 16.6iT - 361T^{2} \)
23 \( 1 + (6.65 + 6.65i)T + 529iT^{2} \)
29 \( 1 + 0.0454iT - 841T^{2} \)
31 \( 1 - 4.49T + 961T^{2} \)
37 \( 1 + (35.3 - 35.3i)T - 1.36e3iT^{2} \)
41 \( 1 + 20.2T + 1.68e3T^{2} \)
43 \( 1 + (32.2 + 32.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (50.5 - 50.5i)T - 2.20e3iT^{2} \)
53 \( 1 + (5.50 + 5.50i)T + 2.80e3iT^{2} \)
59 \( 1 - 55.4iT - 3.48e3T^{2} \)
61 \( 1 - 47.8T + 3.72e3T^{2} \)
67 \( 1 + (-85.2 + 85.2i)T - 4.48e3iT^{2} \)
71 \( 1 + 48.4T + 5.04e3T^{2} \)
73 \( 1 + (-21.9 - 21.9i)T + 5.32e3iT^{2} \)
79 \( 1 - 126. iT - 6.24e3T^{2} \)
83 \( 1 + (-94.9 - 94.9i)T + 6.88e3iT^{2} \)
89 \( 1 + 71.7iT - 7.92e3T^{2} \)
97 \( 1 + (-37 + 37i)T - 9.40e3iT^{2} \)
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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

−8.464605862622837035395016275762, −8.028371878410979278516505193698, −7.01236431715650850267505961758, −6.46511046855768946764695522765, −5.21794485582512507503253009708, −4.57631691218532036584591948955, −3.71159077621917873505750563959, −2.48431590773195886236978996586, −1.39276645989792265415420596736, −0.04976566118830241009351476438, 1.75463237958958950307720057065, 2.37805854306248123029111828891, 3.74537719624819617674944962489, 4.77165234115165595819121431820, 5.24701295621894428009259561947, 6.48906939441477300097697302717, 7.04343881513526714319806500649, 8.002112849147330140089080843819, 8.844623228370468447664449880912, 9.366023161559868620101063134322

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