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Properties

Label 1-2541-2541.20-r0-0-0
Degree $1$
Conductor $2541$
Sign $-0.372 + 0.928i$
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  + (0.362 + 0.931i)2-s + (−0.736 + 0.676i)4-s + (0.696 + 0.717i)5-s + (−0.897 − 0.441i)8-s + (−0.415 + 0.909i)10-s + (−0.198 + 0.980i)13-s + (0.0855 − 0.996i)16-s + (0.610 − 0.791i)17-s + (0.564 − 0.825i)19-s + (−0.998 − 0.0570i)20-s + (0.654 − 0.755i)23-s + (−0.0285 + 0.999i)25-s + (−0.985 + 0.170i)26-s + (0.0285 + 0.999i)29-s + (0.870 − 0.491i)31-s + (0.959 − 0.281i)32-s + ⋯
L(s)  = 1  + (0.362 + 0.931i)2-s + (−0.736 + 0.676i)4-s + (0.696 + 0.717i)5-s + (−0.897 − 0.441i)8-s + (−0.415 + 0.909i)10-s + (−0.198 + 0.980i)13-s + (0.0855 − 0.996i)16-s + (0.610 − 0.791i)17-s + (0.564 − 0.825i)19-s + (−0.998 − 0.0570i)20-s + (0.654 − 0.755i)23-s + (−0.0285 + 0.999i)25-s + (−0.985 + 0.170i)26-s + (0.0285 + 0.999i)29-s + (0.870 − 0.491i)31-s + (0.959 − 0.281i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
Sign: $-0.372 + 0.928i$
Analytic conductor: \(11.8003\)
Root analytic conductor: \(11.8003\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2541} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2541,\ (0:\ ),\ -0.372 + 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.263701252 + 1.868858710i\)
\(L(\frac12)\) \(\approx\) \(1.263701252 + 1.868858710i\)
\(L(1)\) \(\approx\) \(1.115250879 + 0.8521285393i\)
\(L(1)\) \(\approx\) \(1.115250879 + 0.8521285393i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.362 + 0.931i)T \)
5 \( 1 + (0.696 + 0.717i)T \)
13 \( 1 + (-0.198 + 0.980i)T \)
17 \( 1 + (0.610 - 0.791i)T \)
19 \( 1 + (0.564 - 0.825i)T \)
23 \( 1 + (0.654 - 0.755i)T \)
29 \( 1 + (0.0285 + 0.999i)T \)
31 \( 1 + (0.870 - 0.491i)T \)
37 \( 1 + (0.993 + 0.113i)T \)
41 \( 1 + (0.774 - 0.633i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + (-0.254 - 0.967i)T \)
53 \( 1 + (-0.0855 - 0.996i)T \)
59 \( 1 + (0.774 + 0.633i)T \)
61 \( 1 + (0.362 - 0.931i)T \)
67 \( 1 + (0.841 + 0.540i)T \)
71 \( 1 + (-0.941 + 0.336i)T \)
73 \( 1 + (0.921 + 0.389i)T \)
79 \( 1 + (-0.466 + 0.884i)T \)
83 \( 1 + (0.974 + 0.226i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (-0.696 + 0.717i)T \)
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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

−19.365486185387845440850736335781, −18.6285194296544770439969974697, −17.70378647886322387615727422001, −17.370949721522602596773111695286, −16.46077154046372894589171045113, −15.480500491601132359148603532040, −14.75806981816328181688823159649, −13.97893268182604459573121460768, −13.33960316610392758363757572398, −12.68291173670005411466586647801, −12.18867570891689438970724389544, −11.33012653551276650838633503686, −10.389172761053196519441025003624, −9.90383399185025713187078318640, −9.27852034161280046910499901225, −8.34069112645350390370180745515, −7.708176348152683214455412636124, −6.18364637244787690843988718517, −5.70839393685886424338058153817, −5.009635183744649677571158980350, −4.166113795534623961168339591129, −3.25631423554899733710432666178, −2.47496510334387787064082682614, −1.45568085466798253927993079297, −0.84754272774974670813418444837, 0.95009255453268386517771279798, 2.42691576684961197553407528868, 3.00701007981683010708721171439, 4.04101113290407922975885905418, 4.96696955780011928413133309862, 5.51791299964490098580089981527, 6.64790192171359660594400084606, 6.82287934570842516355680948696, 7.69275344503510616298509336423, 8.6747047731240085238139634797, 9.46698371960729589859725391851, 9.903710960784918629214096848006, 11.12319657652490690647412252707, 11.73061772428741827112089108193, 12.740095487825757596843556285520, 13.40752735818149305593628242930, 14.14286065998920947823322431290, 14.51935766146785428210719585613, 15.261284562366048480780258657496, 16.192771563396726335405651538214, 16.6672340662649681801217678657, 17.47152890163687525169977754196, 18.140731235986708974363703041222, 18.64888839885331569971545579997, 19.41810207205584986588077115337

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