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Properties

Label 1-2541-2541.188-r0-0-0
Degree $1$
Conductor $2541$
Sign $-0.244 + 0.969i$
Analytic cond. $11.8003$
Root an. cond. $11.8003$
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  + (−0.841 + 0.540i)2-s + (0.415 − 0.909i)4-s + (−0.142 + 0.989i)5-s + (0.142 + 0.989i)8-s + (−0.415 − 0.909i)10-s + (−0.415 + 0.909i)13-s + (−0.654 − 0.755i)16-s + (−0.959 − 0.281i)17-s + (0.959 − 0.281i)19-s + (0.841 + 0.540i)20-s + (0.654 + 0.755i)23-s + (−0.959 − 0.281i)25-s + (−0.142 − 0.989i)26-s + (0.959 − 0.281i)29-s + (−0.415 − 0.909i)31-s + (0.959 + 0.281i)32-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)2-s + (0.415 − 0.909i)4-s + (−0.142 + 0.989i)5-s + (0.142 + 0.989i)8-s + (−0.415 − 0.909i)10-s + (−0.415 + 0.909i)13-s + (−0.654 − 0.755i)16-s + (−0.959 − 0.281i)17-s + (0.959 − 0.281i)19-s + (0.841 + 0.540i)20-s + (0.654 + 0.755i)23-s + (−0.959 − 0.281i)25-s + (−0.142 − 0.989i)26-s + (0.959 − 0.281i)29-s + (−0.415 − 0.909i)31-s + (0.959 + 0.281i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
Sign: $-0.244 + 0.969i$
Analytic conductor: \(11.8003\)
Root analytic conductor: \(11.8003\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2541} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2541,\ (0:\ ),\ -0.244 + 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5924532400 + 0.7601118850i\)
\(L(\frac12)\) \(\approx\) \(0.5924532400 + 0.7601118850i\)
\(L(1)\) \(\approx\) \(0.6533229355 + 0.3123968484i\)
\(L(1)\) \(\approx\) \(0.6533229355 + 0.3123968484i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.841 + 0.540i)T \)
5 \( 1 + (-0.142 + 0.989i)T \)
13 \( 1 + (-0.415 + 0.909i)T \)
17 \( 1 + (-0.959 - 0.281i)T \)
19 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (0.959 - 0.281i)T \)
31 \( 1 + (-0.415 - 0.909i)T \)
37 \( 1 + (0.415 + 0.909i)T \)
41 \( 1 + (0.841 - 0.540i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (0.841 + 0.540i)T \)
53 \( 1 + (0.654 - 0.755i)T \)
59 \( 1 + (0.841 + 0.540i)T \)
61 \( 1 + (-0.841 - 0.540i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (0.654 + 0.755i)T \)
79 \( 1 + (-0.142 + 0.989i)T \)
83 \( 1 + (-0.654 + 0.755i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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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

−19.5128853687768112061777006150, −18.35446196376710856899033680520, −17.85534668115006365828422418406, −17.21394142123451302396837334645, −16.44932955473455425879225738651, −15.92583430241388709051958716949, −15.20929267354936977741281484991, −14.12350424068164244273324329752, −13.08672166527168466401323536374, −12.6701938866619604972876467625, −12.04618951757576247463726453676, −11.18661973924750063307418333736, −10.49755474118738728098692582469, −9.718066860549774854421909788775, −8.973410878295200691841511742901, −8.435973852117963951881027793658, −7.683361843030098878806925383083, −6.9545744317065520185641484244, −5.87493700542543679883151798833, −4.910738417256543007478581571461, −4.16808970828240445089699791835, −3.16039926260093553257667147403, −2.37583541826724856662546239982, −1.29796506921479372378506708208, −0.56559480360305926865908473593, 0.85740575707127442876674554591, 2.12348982812969098988983897458, 2.671193206029101857690836881875, 3.874782804480265663439197664310, 4.879336878763658525140081194736, 5.759910979876605055696003246879, 6.683943184129987736403195051144, 7.0818543484722209631356113456, 7.76561919946086898663020401793, 8.71866816061553224471494345704, 9.52078447141440287738614702204, 9.96055504590506874044611096752, 11.11851362008925786760361640366, 11.272470449193412796204183898291, 12.185205207830036566211061872641, 13.63551735714199270293357372373, 13.94480614317236381241747734826, 14.85658093331889256190063356693, 15.46830046925089488226987145630, 15.96342404758981892573741303194, 16.96757885851403614727589465316, 17.478479761851960612323719283897, 18.30454483040880553029738288175, 18.71030961222748141065378368129, 19.53116578226004762722077944281

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