↑

Properties

Label 2-15e2-45.32-c1-0-10
Degree $2$
Conductor $225$
Sign $-0.422 + 0.906i$
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  + (−1.29 − 0.347i)2-s + (1.25 − 1.18i)3-s + (−0.170 − 0.0981i)4-s + (−2.04 + 1.10i)6-s + (0.530 − 1.97i)7-s + (2.08 + 2.08i)8-s + (0.170 − 2.99i)9-s + (−0.762 + 0.440i)11-s + (−0.330 + 0.0786i)12-s + (−1.43 − 5.36i)13-s + (−1.37 + 2.38i)14-s + (−1.78 − 3.09i)16-s + (−1.13 + 1.13i)17-s + (−1.26 + 3.82i)18-s + 1.52i·19-s + ⋯
L(s)  = 1  + (−0.917 − 0.245i)2-s + (0.726 − 0.686i)3-s + (−0.0850 − 0.0490i)4-s + (−0.835 + 0.451i)6-s + (0.200 − 0.747i)7-s + (0.737 + 0.737i)8-s + (0.0566 − 0.998i)9-s + (−0.229 + 0.132i)11-s + (−0.0955 + 0.0227i)12-s + (−0.398 − 1.48i)13-s + (−0.367 + 0.636i)14-s + (−0.446 − 0.772i)16-s + (−0.275 + 0.275i)17-s + (−0.297 + 0.901i)18-s + 0.349i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.422 + 0.906i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.422 + 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.447469 - 0.701934i\)
\(L(\frac12)\) \(\approx\) \(0.447469 - 0.701934i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.25 + 1.18i)T \)
5 \( 1 \)
good2 \( 1 + (1.29 + 0.347i)T + (1.73 + i)T^{2} \)
7 \( 1 + (-0.530 + 1.97i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.762 - 0.440i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.43 + 5.36i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (1.13 - 1.13i)T - 17iT^{2} \)
19 \( 1 - 1.52iT - 19T^{2} \)
23 \( 1 + (1.53 - 0.410i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.796 + 1.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.49 + 6.05i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.25 - 4.25i)T + 37iT^{2} \)
41 \( 1 + (-3.11 - 1.79i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.85 - 0.497i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-7.99 - 2.14i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.65 - 4.65i)T + 53iT^{2} \)
59 \( 1 + (3.81 - 6.61i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.64 - 11.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.20 - 0.859i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 5.89iT - 71T^{2} \)
73 \( 1 + (1.58 - 1.58i)T - 73iT^{2} \)
79 \( 1 + (-6.69 + 3.86i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.57 + 9.59i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 4.62T + 89T^{2} \)
97 \( 1 + (-1.02 + 3.82i)T + (-84.0 - 48.5i)T^{2} \)
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

−11.89261184970672837097932684075, −10.60989269986134535888483301624, −9.984127586606554307431866716961, −8.943200040179217747957241901991, −7.86966827017060533356484957375, −7.55700794452894087565423164208, −5.88042834841103205573247244788, −4.25918314280505860439919673547, −2.55151502871548481856757504841, −0.914490822482773851608930813121, 2.27040980463795434685399641499, 3.99144442924138018855191033328, 5.05626691255796274167934315892, 6.84403814886292156270868073784, 7.920747573074502283875048478966, 8.899536230952795217412979111542, 9.264207377480700331476849672652, 10.27485714084100189804712853154, 11.33542397987183612585400035068, 12.54547715434090777732264254881

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