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Properties

Label 2-15e2-45.34-c1-0-14
Degree $2$
Conductor $225$
Sign $-0.232 + 0.972i$
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  + (0.866 + 0.5i)2-s + (−0.866 − 1.5i)3-s + (−0.500 − 0.866i)4-s − 1.73i·6-s + (−2.59 − 1.5i)7-s − 3i·8-s + (−1.5 + 2.59i)9-s + (1 − 1.73i)11-s + (−0.866 + 1.5i)12-s + (−1.73 + i)13-s + (−1.5 − 2.59i)14-s + (0.500 − 0.866i)16-s − 4i·17-s + (−2.59 + 1.5i)18-s + 8·19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 − 0.866i)3-s + (−0.250 − 0.433i)4-s − 0.707i·6-s + (−0.981 − 0.566i)7-s − 1.06i·8-s + (−0.5 + 0.866i)9-s + (0.301 − 0.522i)11-s + (−0.249 + 0.433i)12-s + (−0.480 + 0.277i)13-s + (−0.400 − 0.694i)14-s + (0.125 − 0.216i)16-s − 0.970i·17-s + (−0.612 + 0.353i)18-s + 1.83·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.232 + 0.972i$
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: \((2,\ 225,\ (\ :1/2),\ -0.232 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.663235 - 0.840329i\)
\(L(\frac12)\) \(\approx\) \(0.663235 - 0.840329i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 1.5i)T \)
5 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (2.59 + 1.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.73 - i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 + (-2.59 + 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.92 - 4i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.06 - 3.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 + (7 + 12.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.59 + 1.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + (3 - 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.79 + 4.5i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 + (-1.73 - i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.20249156234456946547407791098, −11.20175042978215047880288400662, −10.00654806075417124756598694700, −9.169397830076688743504235621598, −7.43938691140677565904046477327, −6.79013835701274226589800914758, −5.81671718120177359628184083453, −4.82443299653086056644195378616, −3.20299101371822093947110918609, −0.815525097298749108345657997160, 2.92472370356700460003154624998, 3.87077696061692530105290520323, 5.08743014639196802501939699801, 5.97373834221418650207816815852, 7.47395685017942830436648964286, 8.973711634705946568150305396738, 9.602821310844246444855257997651, 10.69425878423511460153382332001, 11.87981110438293629006701230438, 12.29760985805107917983992249886

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