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Properties

Label 2-1823-1823.1822-c0-0-0
Degree $2$
Conductor $1823$
Sign $-1$
Analytic cond. $0.909795$
Root an. cond. $0.953832$
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  + 2-s − 3-s + 1.41i·5-s − 6-s − 8-s + 1.41i·10-s − 1.41i·15-s − 16-s − 17-s − 19-s − 1.41i·23-s + 24-s − 1.00·25-s + 27-s − 29-s − 1.41i·30-s + ⋯
L(s)  = 1  + 2-s − 3-s + 1.41i·5-s − 6-s − 8-s + 1.41i·10-s − 1.41i·15-s − 16-s − 17-s − 19-s − 1.41i·23-s + 24-s − 1.00·25-s + 27-s − 29-s − 1.41i·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1823 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1823 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1823\)
Sign: $-1$
Analytic conductor: \(0.909795\)
Root analytic conductor: \(0.953832\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1823} (1822, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1823,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3287947163\)
\(L(\frac12)\) \(\approx\) \(0.3287947163\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1823 \( 1 + T \)
good2 \( 1 - T + T^{2} \)
3 \( 1 + T + T^{2} \)
5 \( 1 - 1.41iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 - 1.41iT - T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 1.41iT - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.19426338694830651050468217017, −9.065473616697837102429454544391, −8.255657503545368210913902388041, −6.92149119690920107446060885238, −6.45531127710601782182871403257, −5.94350529921170722447252196163, −4.93113734725022865665633071842, −4.23684216128027213963907488396, −3.18496409858274787511975126566, −2.36304402102288405314046745953, 0.18779014862376077482994434129, 1.88852319838229760156021709377, 3.48841890972992842946176303250, 4.39581639714910822412826127356, 4.99878985758791798383242189449, 5.60901660620309383578252196875, 6.20552075336812835245650300044, 7.25627932497568030731022628683, 8.596127298170167728240007798773, 8.870712269563117716579047082235

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