Properties

Label 1-2243-2243.2242-r1-0-0
Degree $1$
Conductor $2243$
Sign $1$
Analytic cond. $241.043$
Root an. cond. $241.043$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 13-s − 14-s − 15-s + 16-s + 17-s − 18-s − 19-s − 20-s + 21-s − 22-s − 23-s − 24-s + 25-s + 26-s + 27-s + 28-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 13-s − 14-s − 15-s + 16-s + 17-s − 18-s − 19-s − 20-s + 21-s − 22-s − 23-s − 24-s + 25-s + 26-s + 27-s + 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2243 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2243 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2243\)
Sign: $1$
Analytic conductor: \(241.043\)
Root analytic conductor: \(241.043\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2243} (2242, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 2243,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.038040780\)
\(L(\frac12)\) \(\approx\) \(2.038040780\)
\(L(1)\) \(\approx\) \(0.9950078222\)
\(L(1)\) \(\approx\) \(0.9950078222\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2243 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 - T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.43186918791939500101548540219, −18.98806972041513274553296357247, −18.295989043525027120939395838143, −17.26745055346793703035409631813, −16.82169230166783720184320102289, −15.850097846231549904628422125117, −15.15603291317585717655963556084, −14.59892162106918067565026612526, −14.17172785447573241609591630379, −12.710900836004084595798699718656, −11.99238677343877499692057093934, −11.5634618096830647103509540880, −10.484976006332821769375556820345, −9.90080791284367163494284100616, −8.933466626925410767099448747606, −8.415666459812833325448180170322, −7.74286332643166727000899989681, −7.28785856102665457777796141997, −6.39288991455591189969694704900, −5.03087550966584695031761010787, −4.06708919937852248407479876445, −3.40580544465005398330573094149, −2.29910020335818504963359147060, −1.6340477659251747740384955327, −0.63627918597736951321476328381, 0.63627918597736951321476328381, 1.6340477659251747740384955327, 2.29910020335818504963359147060, 3.40580544465005398330573094149, 4.06708919937852248407479876445, 5.03087550966584695031761010787, 6.39288991455591189969694704900, 7.28785856102665457777796141997, 7.74286332643166727000899989681, 8.415666459812833325448180170322, 8.933466626925410767099448747606, 9.90080791284367163494284100616, 10.484976006332821769375556820345, 11.5634618096830647103509540880, 11.99238677343877499692057093934, 12.710900836004084595798699718656, 14.17172785447573241609591630379, 14.59892162106918067565026612526, 15.15603291317585717655963556084, 15.850097846231549904628422125117, 16.82169230166783720184320102289, 17.26745055346793703035409631813, 18.295989043525027120939395838143, 18.98806972041513274553296357247, 19.43186918791939500101548540219

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