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Properties

Label 2-15e2-225.106-c1-0-25
Degree $2$
Conductor $225$
Sign $-0.880 + 0.473i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.745 − 0.158i)2-s + (1.13 − 1.30i)3-s + (−1.29 − 0.576i)4-s + (−1.26 − 1.84i)5-s + (−1.05 + 0.793i)6-s + (−0.476 + 0.825i)7-s + (2.10 + 1.53i)8-s + (−0.409 − 2.97i)9-s + (0.650 + 1.57i)10-s + (−2.83 − 0.601i)11-s + (−2.22 + 1.03i)12-s + (−1.91 + 0.406i)13-s + (0.486 − 0.540i)14-s + (−3.84 − 0.449i)15-s + (0.567 + 0.630i)16-s + (−5.51 − 4.01i)17-s + ⋯
L(s)  = 1  + (−0.527 − 0.112i)2-s + (0.657 − 0.753i)3-s + (−0.647 − 0.288i)4-s + (−0.565 − 0.824i)5-s + (−0.431 + 0.323i)6-s + (−0.180 + 0.312i)7-s + (0.745 + 0.541i)8-s + (−0.136 − 0.990i)9-s + (0.205 + 0.498i)10-s + (−0.853 − 0.181i)11-s + (−0.643 + 0.298i)12-s + (−0.530 + 0.112i)13-s + (0.130 − 0.144i)14-s + (−0.993 − 0.116i)15-s + (0.141 + 0.157i)16-s + (−1.33 − 0.972i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.880 + 0.473i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.880 + 0.473i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.160026 - 0.635300i\)
\(L(\frac12)\) \(\approx\) \(0.160026 - 0.635300i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.13 + 1.30i)T \)
5 \( 1 + (1.26 + 1.84i)T \)
good2 \( 1 + (0.745 + 0.158i)T + (1.82 + 0.813i)T^{2} \)
7 \( 1 + (0.476 - 0.825i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.83 + 0.601i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (1.91 - 0.406i)T + (11.8 - 5.28i)T^{2} \)
17 \( 1 + (5.51 + 4.01i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-3.77 - 2.74i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-5.89 + 6.54i)T + (-2.40 - 22.8i)T^{2} \)
29 \( 1 + (0.0975 + 0.928i)T + (-28.3 + 6.02i)T^{2} \)
31 \( 1 + (-0.277 + 2.64i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (-0.799 + 2.46i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-5.30 + 1.12i)T + (37.4 - 16.6i)T^{2} \)
43 \( 1 + (-4.43 + 7.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.419 - 3.99i)T + (-45.9 + 9.77i)T^{2} \)
53 \( 1 + (-4.30 + 3.12i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.69 - 0.997i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (-10.9 - 2.33i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (0.387 - 3.68i)T + (-65.5 - 13.9i)T^{2} \)
71 \( 1 + (0.125 - 0.0911i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.09 + 9.53i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.442 + 4.20i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (3.01 - 1.34i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (-5.32 - 16.3i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-0.203 - 1.93i)T + (-94.8 + 20.1i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.01155487731149944145004287074, −10.83050794779124010974024526142, −9.425124444192361685372980955751, −8.937778046438936715161841679092, −8.052292577505762042985797977938, −7.17851166249489890994395311894, −5.47982708702887759089406558216, −4.34342394276904028147488471374, −2.51563127147468921151787056141, −0.60824170508875731731205220293, 2.86527859395438933830824810601, 3.97855418065213926592295379950, 5.04695200678430101787393261032, 7.10057029693469666370191608582, 7.77764016253405570664529625420, 8.782711687410716372317984966431, 9.695589290431775679572548868852, 10.51480411632969863068955754749, 11.29856167518124727904464625385, 12.95587017068184048734590971316

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