Properties

Label 2-15e2-225.16-c1-0-12
Degree $2$
Conductor $225$
Sign $0.988 + 0.153i$
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  + (0.520 − 0.231i)2-s + (−1.62 − 0.597i)3-s + (−1.12 + 1.24i)4-s + (2.21 + 0.336i)5-s + (−0.984 + 0.0659i)6-s + (1.58 − 2.75i)7-s + (−0.646 + 1.99i)8-s + (2.28 + 1.94i)9-s + (1.22 − 0.336i)10-s + (1.73 − 0.772i)11-s + (2.56 − 1.35i)12-s + (5.02 + 2.23i)13-s + (0.189 − 1.79i)14-s + (−3.39 − 1.86i)15-s + (−0.225 − 2.14i)16-s + (0.888 − 2.73i)17-s + ⋯
L(s)  = 1  + (0.367 − 0.163i)2-s + (−0.938 − 0.344i)3-s + (−0.560 + 0.622i)4-s + (0.988 + 0.150i)5-s + (−0.401 + 0.0269i)6-s + (0.600 − 1.04i)7-s + (−0.228 + 0.703i)8-s + (0.762 + 0.647i)9-s + (0.388 − 0.106i)10-s + (0.522 − 0.232i)11-s + (0.740 − 0.391i)12-s + (1.39 + 0.620i)13-s + (0.0505 − 0.481i)14-s + (−0.876 − 0.482i)15-s + (−0.0564 − 0.536i)16-s + (0.215 − 0.663i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.988 + 0.153i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.988 + 0.153i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25204 - 0.0969183i\)
\(L(\frac12)\) \(\approx\) \(1.25204 - 0.0969183i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.62 + 0.597i)T \)
5 \( 1 + (-2.21 - 0.336i)T \)
good2 \( 1 + (-0.520 + 0.231i)T + (1.33 - 1.48i)T^{2} \)
7 \( 1 + (-1.58 + 2.75i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.73 + 0.772i)T + (7.36 - 8.17i)T^{2} \)
13 \( 1 + (-5.02 - 2.23i)T + (8.69 + 9.66i)T^{2} \)
17 \( 1 + (-0.888 + 2.73i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.39 - 4.27i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.335 + 3.19i)T + (-22.4 - 4.78i)T^{2} \)
29 \( 1 + (1.75 + 0.373i)T + (26.4 + 11.7i)T^{2} \)
31 \( 1 + (7.35 - 1.56i)T + (28.3 - 12.6i)T^{2} \)
37 \( 1 + (4.57 - 3.32i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (0.974 + 0.433i)T + (27.4 + 30.4i)T^{2} \)
43 \( 1 + (3.64 - 6.31i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.34 - 1.56i)T + (42.9 + 19.1i)T^{2} \)
53 \( 1 + (0.982 + 3.02i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (12.0 + 5.37i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (-0.958 + 0.426i)T + (40.8 - 45.3i)T^{2} \)
67 \( 1 + (-12.7 + 2.70i)T + (61.2 - 27.2i)T^{2} \)
71 \( 1 + (2.10 + 6.47i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.173 + 0.126i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.79 - 1.01i)T + (72.1 + 32.1i)T^{2} \)
83 \( 1 + (10.0 + 11.1i)T + (-8.67 + 82.5i)T^{2} \)
89 \( 1 + (2.15 + 1.56i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (13.1 + 2.80i)T + (88.6 + 39.4i)T^{2} \)
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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

−12.29717532518352056734643434561, −11.24683070826857424988527378820, −10.63215259580686615269060020341, −9.367425557844263193112602085257, −8.195787867845575138787315908298, −6.99081139696808526970550767241, −5.97562350673640946340988760026, −4.83175221657701219981128042851, −3.73762530490596195729281441676, −1.52714267270283330888205751768, 1.50486351016520498090083715689, 3.97207795363037206962742928310, 5.42867793057593226971397096853, 5.60552499423282267606264464956, 6.68499757859594113039581892027, 8.750788992574316575244613690910, 9.316668161326972138681151945006, 10.45339209300123450145906801728, 11.16569431211523118887819322205, 12.42936911694346334652783170716

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