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Properties

Label 1-15e2-225.14-r1-0-0
Degree $1$
Conductor $225$
Sign $-0.752 - 0.658i$
Analytic cond. $24.1796$
Root an. cond. $24.1796$
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  + (0.669 + 0.743i)2-s + (−0.104 + 0.994i)4-s + (0.5 − 0.866i)7-s + (−0.809 + 0.587i)8-s + (−0.669 − 0.743i)11-s + (−0.669 + 0.743i)13-s + (0.978 − 0.207i)14-s + (−0.978 − 0.207i)16-s + (−0.809 + 0.587i)17-s + (−0.809 + 0.587i)19-s + (0.104 − 0.994i)22-s + (−0.978 + 0.207i)23-s − 26-s + (0.809 + 0.587i)28-s + (−0.913 + 0.406i)29-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)2-s + (−0.104 + 0.994i)4-s + (0.5 − 0.866i)7-s + (−0.809 + 0.587i)8-s + (−0.669 − 0.743i)11-s + (−0.669 + 0.743i)13-s + (0.978 − 0.207i)14-s + (−0.978 − 0.207i)16-s + (−0.809 + 0.587i)17-s + (−0.809 + 0.587i)19-s + (0.104 − 0.994i)22-s + (−0.978 + 0.207i)23-s − 26-s + (0.809 + 0.587i)28-s + (−0.913 + 0.406i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.752 - 0.658i$
Analytic conductor: \(24.1796\)
Root analytic conductor: \(24.1796\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (1:\ ),\ -0.752 - 0.658i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1544617391 + 0.4109387974i\)
\(L(\frac12)\) \(\approx\) \(-0.1544617391 + 0.4109387974i\)
\(L(1)\) \(\approx\) \(0.9421914363 + 0.4857700989i\)
\(L(1)\) \(\approx\) \(0.9421914363 + 0.4857700989i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.669 + 0.743i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (-0.669 + 0.743i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.978 + 0.207i)T \)
29 \( 1 + (-0.913 + 0.406i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (-0.669 + 0.743i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.913 - 0.406i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.669 + 0.743i)T \)
61 \( 1 + (0.669 + 0.743i)T \)
67 \( 1 + (-0.913 - 0.406i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (0.913 - 0.406i)T \)
83 \( 1 + (-0.104 - 0.994i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (-0.913 + 0.406i)T \)
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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

−25.431311829202547984946701333783, −24.46999938924147214674466362354, −23.768077568878454358120613242598, −22.51188222959313925751825992609, −22.0514782200303059122961645892, −20.882561601706554790028996656398, −20.28750503232134076897808464387, −19.16926607041313753080148383762, −18.23158813297570957681707513607, −17.42631718082113250788380452860, −15.47666828549093183960252231520, −15.25906932781926061758114065897, −14.01547137668979163581096057109, −12.92233906948867501234300354623, −12.18143809661250333866244849011, −11.201538140106674951537747552222, −10.171545335657401805764274069285, −9.16884580787637245339335761277, −7.84798089534665458039073302980, −6.349423469709108852035178778572, −5.1784284501206205370487974333, −4.4417181698712615839894014682, −2.7372820512591957133445400734, −2.034777886266720976134438306337, −0.10279064519077877670031278448, 2.11818511839064188836231402439, 3.72245382907657653188365734876, 4.57228655740312965203309818551, 5.77961035847203730270884941075, 6.88993188735585631830285013917, 7.868735730111150416511479542662, 8.767776514424444294092364761053, 10.36954114953931466376762080591, 11.4169109593066993496186131056, 12.57183373487076394824938647605, 13.6043097630619264322861149809, 14.27325079360402670229141295749, 15.2825232489715328664989931659, 16.37773989147004618901731833265, 17.07875419286064934642556179909, 18.00530465874441396992052778627, 19.29283978674609076735724061105, 20.51039532566576654390819050160, 21.38393420767430741379213826756, 22.12961833882759093291760793298, 23.37417050349232463899328545601, 23.937718306805203220303043912018, 24.619681452285274017275105946782, 25.94665865002845112933144330128, 26.55027106882925748899793889983

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