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Properties

Label 2-15e2-225.167-c1-0-26
Degree $2$
Conductor $225$
Sign $-0.982 - 0.188i$
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  + (−0.113 − 2.16i)2-s + (0.935 − 1.45i)3-s + (−2.67 + 0.280i)4-s + (−0.996 − 2.00i)5-s + (−3.25 − 1.85i)6-s + (−0.983 + 3.67i)7-s + (0.231 + 1.46i)8-s + (−1.24 − 2.72i)9-s + (−4.21 + 2.38i)10-s + (−0.855 − 0.769i)11-s + (−2.08 + 4.15i)12-s + (4.58 + 0.240i)13-s + (8.04 + 1.71i)14-s + (−3.85 − 0.420i)15-s + (−2.11 + 0.449i)16-s + (0.222 − 0.0351i)17-s + ⋯
L(s)  = 1  + (−0.0801 − 1.52i)2-s + (0.540 − 0.841i)3-s + (−1.33 + 0.140i)4-s + (−0.445 − 0.895i)5-s + (−1.32 − 0.758i)6-s + (−0.371 + 1.38i)7-s + (0.0820 + 0.517i)8-s + (−0.416 − 0.909i)9-s + (−1.33 + 0.752i)10-s + (−0.257 − 0.232i)11-s + (−0.603 + 1.19i)12-s + (1.27 + 0.0666i)13-s + (2.15 + 0.457i)14-s + (−0.994 − 0.108i)15-s + (−0.528 + 0.112i)16-s + (0.0538 − 0.00852i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.982 - 0.188i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.982 - 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.110311 + 1.16017i\)
\(L(\frac12)\) \(\approx\) \(0.110311 + 1.16017i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.935 + 1.45i)T \)
5 \( 1 + (0.996 + 2.00i)T \)
good2 \( 1 + (0.113 + 2.16i)T + (-1.98 + 0.209i)T^{2} \)
7 \( 1 + (0.983 - 3.67i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.855 + 0.769i)T + (1.14 + 10.9i)T^{2} \)
13 \( 1 + (-4.58 - 0.240i)T + (12.9 + 1.35i)T^{2} \)
17 \( 1 + (-0.222 + 0.0351i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (-4.93 + 6.79i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.558 + 0.859i)T + (-9.35 + 21.0i)T^{2} \)
29 \( 1 + (-7.73 - 3.44i)T + (19.4 + 21.5i)T^{2} \)
31 \( 1 + (-0.366 + 0.163i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (2.61 - 1.33i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (-1.49 + 1.34i)T + (4.28 - 40.7i)T^{2} \)
43 \( 1 + (3.34 + 0.896i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (4.96 - 1.90i)T + (34.9 - 31.4i)T^{2} \)
53 \( 1 + (-7.49 - 1.18i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (3.36 + 3.74i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (-7.40 + 8.22i)T + (-6.37 - 60.6i)T^{2} \)
67 \( 1 + (-11.9 - 4.58i)T + (49.7 + 44.8i)T^{2} \)
71 \( 1 + (1.92 + 2.64i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-5.69 - 2.90i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (3.96 - 8.89i)T + (-52.8 - 58.7i)T^{2} \)
83 \( 1 + (9.36 - 7.58i)T + (17.2 - 81.1i)T^{2} \)
89 \( 1 + (0.0846 + 0.260i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-1.76 - 4.60i)T + (-72.0 + 64.9i)T^{2} \)
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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

−11.82683994192164838562238754650, −11.22192684442979405336503767402, −9.624439330116178156514827219707, −8.840652123600248965942641998371, −8.364631324050207654834548699422, −6.64846555289349255545613793523, −5.19667243633713246218756075182, −3.51765849387376648259059766585, −2.54850701026653705517683177821, −1.03492181316860213162683269106, 3.38426023937230765573956285659, 4.27116102264330745160070425130, 5.76689964918516655255962318579, 6.86685492089294192209811275881, 7.75994273229297973117716718547, 8.388309044372951561796077175005, 9.877237171112268199943224351813, 10.43617669561836340461407263121, 11.55307967696178525963549503300, 13.47524630334256624224022666962

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