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Properties

Label 2-15e2-45.34-c1-0-15
Degree $2$
Conductor $225$
Sign $-0.641 - 0.767i$
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  + (−1.27 − 0.736i)2-s + (−0.350 − 1.69i)3-s + (0.0852 + 0.147i)4-s + (−0.802 + 2.42i)6-s + (−3.34 − 1.93i)7-s + 2.69i·8-s + (−2.75 + 1.18i)9-s + (−0.130 + 0.225i)11-s + (0.220 − 0.196i)12-s + (3.53 − 2.03i)13-s + (2.84 + 4.93i)14-s + (2.15 − 3.73i)16-s + 3.26i·17-s + (4.38 + 0.513i)18-s − 4.24·19-s + ⋯
L(s)  = 1  + (−0.902 − 0.520i)2-s + (−0.202 − 0.979i)3-s + (0.0426 + 0.0738i)4-s + (−0.327 + 0.988i)6-s + (−1.26 − 0.730i)7-s + 0.952i·8-s + (−0.918 + 0.395i)9-s + (−0.0392 + 0.0679i)11-s + (0.0636 − 0.0566i)12-s + (0.979 − 0.565i)13-s + (0.761 + 1.31i)14-s + (0.538 − 0.933i)16-s + 0.790i·17-s + (1.03 + 0.121i)18-s − 0.974·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 - 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.641 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.641 - 0.767i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.641 - 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.103392 + 0.221181i\)
\(L(\frac12)\) \(\approx\) \(0.103392 + 0.221181i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.350 + 1.69i)T \)
5 \( 1 \)
good2 \( 1 + (1.27 + 0.736i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (3.34 + 1.93i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.130 - 0.225i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.53 + 2.03i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.26iT - 17T^{2} \)
19 \( 1 + 4.24T + 19T^{2} \)
23 \( 1 + (7.53 - 4.34i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.11 + 3.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.32 + 2.29i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.27iT - 37T^{2} \)
41 \( 1 + (2.82 + 4.88i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.85 + 4.53i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.23 - 0.714i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.3iT - 53T^{2} \)
59 \( 1 + (3.56 + 6.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.26 - 2.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.77 + 5.64i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.38T + 71T^{2} \)
73 \( 1 + 0.403iT - 73T^{2} \)
79 \( 1 + (1.52 - 2.63i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.96 - 2.29i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.17T + 89T^{2} \)
97 \( 1 + (2.69 + 1.55i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.47172994154290321796927266452, −10.52805530979396097814931235827, −9.884205000658469445340019991203, −8.610892191000188247134929690742, −7.86685820983734553819614374633, −6.55009270634309044718650939246, −5.73881859551080125454188820871, −3.62554731270203079481811661983, −1.91972009416192943948484130411, −0.26800972290765763813557712504, 3.12782956131572896264681802188, 4.32321185851069736503160692505, 6.03453883812204656387331610354, 6.65731945213667473841802991649, 8.362303705196953259186453641129, 8.951235702658935494543820365147, 9.761881464878903777585617339224, 10.51482790071636057457284804395, 11.82299050156739817142807396400, 12.71624555705235473395729855036

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