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Properties

Label 2-2300-460.143-c0-0-1
Degree $2$
Conductor $2300$
Sign $0.877 - 0.479i$
Analytic cond. $1.14784$
Root an. cond. $1.07137$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.479 − 0.877i)2-s + (−0.136 − 1.91i)3-s + (−0.540 + 0.841i)4-s + (−1.61 + 1.03i)6-s + (−1.70 − 0.635i)7-s + (0.997 + 0.0713i)8-s + (−2.65 + 0.381i)9-s + (1.68 + 0.919i)12-s + (0.258 + 1.80i)14-s + (−0.415 − 0.909i)16-s + (1.60 + 2.14i)18-s + (−0.983 + 3.34i)21-s + (−0.877 + 0.479i)23-s − 1.91i·24-s + (0.686 + 3.15i)27-s + (1.45 − 1.09i)28-s + ⋯
L(s)  = 1  + (−0.479 − 0.877i)2-s + (−0.136 − 1.91i)3-s + (−0.540 + 0.841i)4-s + (−1.61 + 1.03i)6-s + (−1.70 − 0.635i)7-s + (0.997 + 0.0713i)8-s + (−2.65 + 0.381i)9-s + (1.68 + 0.919i)12-s + (0.258 + 1.80i)14-s + (−0.415 − 0.909i)16-s + (1.60 + 2.14i)18-s + (−0.983 + 3.34i)21-s + (−0.877 + 0.479i)23-s − 1.91i·24-s + (0.686 + 3.15i)27-s + (1.45 − 1.09i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $0.877 - 0.479i$
Analytic conductor: \(1.14784\)
Root analytic conductor: \(1.07137\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2300} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2300,\ (\ :0),\ 0.877 - 0.479i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07958877258\)
\(L(\frac12)\) \(\approx\) \(0.07958877258\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.479 + 0.877i)T \)
5 \( 1 \)
23 \( 1 + (0.877 - 0.479i)T \)
good3 \( 1 + (0.136 + 1.91i)T + (-0.989 + 0.142i)T^{2} \)
7 \( 1 + (1.70 + 0.635i)T + (0.755 + 0.654i)T^{2} \)
11 \( 1 + (0.841 - 0.540i)T^{2} \)
13 \( 1 + (0.755 - 0.654i)T^{2} \)
17 \( 1 + (0.909 + 0.415i)T^{2} \)
19 \( 1 + (-0.415 - 0.909i)T^{2} \)
29 \( 1 + (0.153 + 0.239i)T + (-0.415 + 0.909i)T^{2} \)
31 \( 1 + (0.142 - 0.989i)T^{2} \)
37 \( 1 + (-0.281 - 0.959i)T^{2} \)
41 \( 1 + (-0.118 + 0.822i)T + (-0.959 - 0.281i)T^{2} \)
43 \( 1 + (-1.07 + 0.0771i)T + (0.989 - 0.142i)T^{2} \)
47 \( 1 + (0.201 + 0.201i)T + iT^{2} \)
53 \( 1 + (0.755 + 0.654i)T^{2} \)
59 \( 1 + (-0.654 - 0.755i)T^{2} \)
61 \( 1 + (0.817 - 0.708i)T + (0.142 - 0.989i)T^{2} \)
67 \( 1 + (1.32 - 0.724i)T + (0.540 - 0.841i)T^{2} \)
71 \( 1 + (-0.841 - 0.540i)T^{2} \)
73 \( 1 + (-0.909 + 0.415i)T^{2} \)
79 \( 1 + (0.654 + 0.755i)T^{2} \)
83 \( 1 + (0.905 - 1.21i)T + (-0.281 - 0.959i)T^{2} \)
89 \( 1 + (-0.368 + 0.425i)T + (-0.142 - 0.989i)T^{2} \)
97 \( 1 + (0.281 - 0.959i)T^{2} \)
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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

−8.488192213338205016630089349621, −7.52450117077408567700706655850, −7.21887038874554582760705581845, −6.35042334614837407348314822753, −5.65767635618092610109684922608, −4.03194246804828890128172051712, −3.11131332898944572386644408924, −2.36430950648966516817976229899, −1.22973417481842185842677688780, −0.06886258897764348418299838648, 2.68741351848520616649352769799, 3.61733784580579957171238779857, 4.41348132092343651744114367284, 5.29061748547303068656992772654, 6.06909882182050124414385660182, 6.41935050923997957357609048689, 7.75809366950007961405124901998, 8.749763931938208396369888001046, 9.200636905580448327405183706761, 9.725447246418685553739813845528

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