↑

Properties

Label 1-4025-4025.33-r1-0-0
Degree $1$
Conductor $4025$
Sign $0.441 + 0.897i$
Analytic cond. $432.546$
Root an. cond. $432.546$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.846 + 0.532i)2-s + (0.917 + 0.398i)3-s + (0.432 + 0.901i)4-s + (0.564 + 0.825i)6-s + (−0.113 + 0.993i)8-s + (0.683 + 0.730i)9-s + (0.969 − 0.244i)11-s + (0.0380 + 0.999i)12-s + (−0.0570 + 0.998i)13-s + (−0.625 + 0.780i)16-s + (−0.0760 − 0.997i)17-s + (0.189 + 0.981i)18-s + (0.851 − 0.524i)19-s + (0.951 + 0.309i)22-s + (−0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (0.846 + 0.532i)2-s + (0.917 + 0.398i)3-s + (0.432 + 0.901i)4-s + (0.564 + 0.825i)6-s + (−0.113 + 0.993i)8-s + (0.683 + 0.730i)9-s + (0.969 − 0.244i)11-s + (0.0380 + 0.999i)12-s + (−0.0570 + 0.998i)13-s + (−0.625 + 0.780i)16-s + (−0.0760 − 0.997i)17-s + (0.189 + 0.981i)18-s + (0.851 − 0.524i)19-s + (0.951 + 0.309i)22-s + (−0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $0.441 + 0.897i$
Analytic conductor: \(432.546\)
Root analytic conductor: \(432.546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (1:\ ),\ 0.441 + 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(6.685969474 + 4.159231641i\)
\(L(\frac12)\) \(\approx\) \(6.685969474 + 4.159231641i\)
\(L(1)\) \(\approx\) \(2.455809056 + 1.237046947i\)
\(L(1)\) \(\approx\) \(2.455809056 + 1.237046947i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.846 + 0.532i)T \)
3 \( 1 + (0.917 + 0.398i)T \)
11 \( 1 + (0.969 - 0.244i)T \)
13 \( 1 + (-0.0570 + 0.998i)T \)
17 \( 1 + (-0.0760 - 0.997i)T \)
19 \( 1 + (0.851 - 0.524i)T \)
29 \( 1 + (0.0285 - 0.999i)T \)
31 \( 1 + (0.595 - 0.803i)T \)
37 \( 1 + (0.730 - 0.683i)T \)
41 \( 1 + (-0.0855 - 0.996i)T \)
43 \( 1 + (0.909 - 0.415i)T \)
47 \( 1 + (-0.207 - 0.978i)T \)
53 \( 1 + (0.986 - 0.161i)T \)
59 \( 1 + (0.548 - 0.836i)T \)
61 \( 1 + (0.749 - 0.662i)T \)
67 \( 1 + (0.938 + 0.345i)T \)
71 \( 1 + (0.897 + 0.441i)T \)
73 \( 1 + (0.983 + 0.179i)T \)
79 \( 1 + (-0.710 + 0.703i)T \)
83 \( 1 + (0.389 - 0.921i)T \)
89 \( 1 + (-0.00951 - 0.999i)T \)
97 \( 1 + (0.389 + 0.921i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.25245654603580609230571625075, −17.815230064617682074706304708582, −16.71052707825737338135640944876, −15.85453415545604182404732205783, −15.12716190981735010142575887326, −14.64359857986135047266355977529, −14.114616109168972059894888994866, −13.38368663812835836347054945038, −12.68046328395542691497530632519, −12.29810300895378577144013753148, −11.50330391790898465156303981539, −10.58270551388675077212375674150, −9.91797284274490990561438188820, −9.307903540981261394774728365326, −8.40290962182322686279913328736, −7.680338262002028149776433043949, −6.803449670502869706230193824779, −6.23008700110715208807368551120, −5.36137965172550890918857002758, −4.410718772555829848724561064886, −3.7127040051607367698306313141, −3.10174165819557469352735899138, −2.38954184691633007475570397792, −1.27601897773007879727067696067, −1.078080369753275506496541022396, 0.765685477783149109541614318839, 2.151155914823818478504364861940, 2.54158357234605979961904542800, 3.69786482098521381396138774043, 3.98778625394149211616873301300, 4.83321792460826254361223845263, 5.560119079629328914991823103039, 6.6244384240129678596922537047, 7.109408715331216371303521704053, 7.85003874556094736356913794419, 8.67504161939786598398883993283, 9.29581678094110861100136754096, 9.85782354268318436409741855792, 11.184916255788985526200607541378, 11.574119602154308644869457202005, 12.338724190898726199355755858076, 13.3701836578302515221009754714, 13.76878596659629022048382791996, 14.30139696493180297092039577009, 14.86649068575293097710757766019, 15.787117837095790668740912872860, 16.025810832954458986920449770092, 16.88845779950615092816070349894, 17.46199322923271663362050013157, 18.565077370344862343509056186958

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