Properties

Label 2-15e2-225.196-c1-0-20
Degree $2$
Conductor $225$
Sign $0.998 - 0.0595i$
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  + (0.510 + 0.566i)2-s + (1.59 + 0.678i)3-s + (0.148 − 1.41i)4-s + (−0.965 − 2.01i)5-s + (0.428 + 1.24i)6-s + (−0.476 + 0.825i)7-s + (2.10 − 1.53i)8-s + (2.07 + 2.16i)9-s + (0.650 − 1.57i)10-s + (1.93 + 2.15i)11-s + (1.19 − 2.14i)12-s + (1.30 − 1.45i)13-s + (−0.711 + 0.151i)14-s + (−0.169 − 3.86i)15-s + (−0.830 − 0.176i)16-s + (−5.51 + 4.01i)17-s + ⋯
L(s)  = 1  + (0.360 + 0.400i)2-s + (0.919 + 0.391i)3-s + (0.0741 − 0.705i)4-s + (−0.431 − 0.901i)5-s + (0.174 + 0.510i)6-s + (−0.180 + 0.312i)7-s + (0.745 − 0.541i)8-s + (0.692 + 0.721i)9-s + (0.205 − 0.498i)10-s + (0.583 + 0.648i)11-s + (0.344 − 0.619i)12-s + (0.362 − 0.402i)13-s + (−0.190 + 0.0403i)14-s + (−0.0437 − 0.999i)15-s + (−0.207 − 0.0441i)16-s + (−1.33 + 0.972i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.998 - 0.0595i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.998 - 0.0595i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84890 + 0.0550648i\)
\(L(\frac12)\) \(\approx\) \(1.84890 + 0.0550648i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.59 - 0.678i)T \)
5 \( 1 + (0.965 + 2.01i)T \)
good2 \( 1 + (-0.510 - 0.566i)T + (-0.209 + 1.98i)T^{2} \)
7 \( 1 + (0.476 - 0.825i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.93 - 2.15i)T + (-1.14 + 10.9i)T^{2} \)
13 \( 1 + (-1.30 + 1.45i)T + (-1.35 - 12.9i)T^{2} \)
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 \( 1 + (8.61 - 1.83i)T + (21.0 - 9.35i)T^{2} \)
29 \( 1 + (-0.852 + 0.379i)T + (19.4 - 21.5i)T^{2} \)
31 \( 1 + (2.42 + 1.08i)T + (20.7 + 23.0i)T^{2} \)
37 \( 1 + (-0.799 - 2.46i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.62 - 4.02i)T + (-4.28 - 40.7i)T^{2} \)
43 \( 1 + (-4.43 + 7.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.66 - 1.63i)T + (31.4 - 34.9i)T^{2} \)
53 \( 1 + (-4.30 - 3.12i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.21 + 3.56i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (7.51 + 8.34i)T + (-6.37 + 60.6i)T^{2} \)
67 \( 1 + (-3.38 - 1.50i)T + (44.8 + 49.7i)T^{2} \)
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 \( 1 + (-3.86 + 1.72i)T + (52.8 - 58.7i)T^{2} \)
83 \( 1 + (-0.344 - 3.28i)T + (-81.1 + 17.2i)T^{2} \)
89 \( 1 + (-5.32 + 16.3i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (1.77 - 0.790i)T + (64.9 - 72.0i)T^{2} \)
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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

−12.52580206075361249912172113546, −11.26613046165776599129453136534, −10.05785901084668174381601708418, −9.291347628550905181184771755429, −8.391728045489494791145700534489, −7.28234437321579066018391422326, −5.96370553565124440578414931603, −4.71305091402669658111443253837, −3.90182398930779587855661943429, −1.82185997593333395987782411206, 2.27449599307561384042546136953, 3.45680498326540726282636963439, 4.14246008105217369892534880385, 6.44345624263073575888410247950, 7.30084236154428334274332250427, 8.161254370121775271344861291811, 9.170995897159525864333114838085, 10.47025500129683857211247631762, 11.58206956227755320984448754065, 12.08739133319940303803769664671

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