↑

Properties

Label 2-15e2-9.7-c1-0-4
Degree $2$
Conductor $225$
Sign $-0.448 - 0.893i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.236 + 0.409i)2-s + (0.544 + 1.64i)3-s + (0.888 + 1.53i)4-s + (−0.802 − 0.165i)6-s + (−1.28 + 2.21i)7-s − 1.78·8-s + (−2.40 + 1.79i)9-s + (3.08 − 5.34i)11-s + (−2.04 + 2.29i)12-s + (−1.06 − 1.84i)13-s + (−0.606 − 1.05i)14-s + (−1.35 + 2.34i)16-s + 3.16·17-s + (−0.164 − 1.41i)18-s + 0.356·19-s + ⋯
L(s)  = 1  + (−0.167 + 0.289i)2-s + (0.314 + 0.949i)3-s + (0.444 + 0.769i)4-s + (−0.327 − 0.0676i)6-s + (−0.484 + 0.838i)7-s − 0.631·8-s + (−0.802 + 0.597i)9-s + (0.929 − 1.61i)11-s + (−0.590 + 0.663i)12-s + (−0.295 − 0.512i)13-s + (−0.162 − 0.280i)14-s + (−0.338 + 0.585i)16-s + 0.768·17-s + (−0.0388 − 0.332i)18-s + 0.0817·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.448 - 0.893i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.448 - 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.663625 + 1.07603i\)
\(L(\frac12)\) \(\approx\) \(0.663625 + 1.07603i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.544 - 1.64i)T \)
5 \( 1 \)
good2 \( 1 + (0.236 - 0.409i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (1.28 - 2.21i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.08 + 5.34i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.06 + 1.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.16T + 17T^{2} \)
19 \( 1 - 0.356T + 19T^{2} \)
23 \( 1 + (-2.10 - 3.64i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.843 - 1.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.12 - 7.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.63T + 37T^{2} \)
41 \( 1 + (-1.36 - 2.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.83 + 6.64i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.71 + 9.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.43T + 53T^{2} \)
59 \( 1 + (5.10 + 8.84i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.00549 + 0.00952i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.491 + 0.851i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.43T + 71T^{2} \)
73 \( 1 - 6.61T + 73T^{2} \)
79 \( 1 + (-4.73 + 8.20i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.20 + 9.02i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.26T + 89T^{2} \)
97 \( 1 + (3.60 - 6.24i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.33964967480733289906571456046, −11.60009858403605093498923483624, −10.65719534332593283810905694822, −9.324141931815932471189040886418, −8.751255168456982648034047940062, −7.83370233661457447654597693714, −6.34931158269681528770240011028, −5.41968076381350467945595907302, −3.58313499514645250464864429052, −2.94377438712154486086497048184, 1.19820753410506600486042892880, 2.55548988144699547985679700709, 4.32497788013079670050577686998, 6.10906914996421960900929565090, 6.89599224443696582073782977687, 7.65406850403249310159064385110, 9.352170793299847023316913259363, 9.798352463633737934587024762937, 11.04821551361582165347361046458, 12.05176871853898066485804074287

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