Properties

Label 2-15e2-225.34-c1-0-6
Degree $2$
Conductor $225$
Sign $0.797 - 0.603i$
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  + (0.368 + 0.826i)2-s + (−1.62 − 0.597i)3-s + (0.790 − 0.877i)4-s + (0.553 + 2.16i)5-s + (−0.104 − 1.56i)6-s + (−0.726 − 0.419i)7-s + (2.73 + 0.889i)8-s + (2.28 + 1.94i)9-s + (−1.58 + 1.25i)10-s + (4.08 − 1.81i)11-s + (−1.80 + 0.954i)12-s + (−0.899 + 2.02i)13-s + (0.0793 − 0.754i)14-s + (0.394 − 3.85i)15-s + (0.0255 + 0.242i)16-s + (6.96 + 2.26i)17-s + ⋯
L(s)  = 1  + (0.260 + 0.584i)2-s + (−0.938 − 0.344i)3-s + (0.395 − 0.438i)4-s + (0.247 + 0.968i)5-s + (−0.0427 − 0.638i)6-s + (−0.274 − 0.158i)7-s + (0.968 + 0.314i)8-s + (0.762 + 0.647i)9-s + (−0.502 + 0.396i)10-s + (1.23 − 0.548i)11-s + (−0.522 + 0.275i)12-s + (−0.249 + 0.560i)13-s + (0.0212 − 0.201i)14-s + (0.101 − 0.994i)15-s + (0.00638 + 0.0607i)16-s + (1.69 + 0.549i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 - 0.603i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.797 - 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.797 - 0.603i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.797 - 0.603i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23204 + 0.413509i\)
\(L(\frac12)\) \(\approx\) \(1.23204 + 0.413509i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.553 - 2.16i)T \)
good2 \( 1 + (-0.368 - 0.826i)T + (-1.33 + 1.48i)T^{2} \)
7 \( 1 + (0.726 + 0.419i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.08 + 1.81i)T + (7.36 - 8.17i)T^{2} \)
13 \( 1 + (0.899 - 2.02i)T + (-8.69 - 9.66i)T^{2} \)
17 \( 1 + (-6.96 - 2.26i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.145 - 0.448i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (7.60 + 0.799i)T + (22.4 + 4.78i)T^{2} \)
29 \( 1 + (3.70 + 0.787i)T + (26.4 + 11.7i)T^{2} \)
31 \( 1 + (-5.21 + 1.10i)T + (28.3 - 12.6i)T^{2} \)
37 \( 1 + (1.78 + 2.45i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (5.98 + 2.66i)T + (27.4 + 30.4i)T^{2} \)
43 \( 1 + (8.54 + 4.93i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.73 - 8.15i)T + (-42.9 - 19.1i)T^{2} \)
53 \( 1 + (-5.86 + 1.90i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-7.09 - 3.16i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (1.71 - 0.763i)T + (40.8 - 45.3i)T^{2} \)
67 \( 1 + (2.49 + 11.7i)T + (-61.2 + 27.2i)T^{2} \)
71 \( 1 + (-1.16 - 3.58i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.42 - 6.09i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (5.43 + 1.15i)T + (72.1 + 32.1i)T^{2} \)
83 \( 1 + (-4.71 + 4.24i)T + (8.67 - 82.5i)T^{2} \)
89 \( 1 + (4.63 + 3.36i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (0.634 - 2.98i)T + (-88.6 - 39.4i)T^{2} \)
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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

−12.02218270376509398652156692948, −11.54139787089713737837478015343, −10.37611776187997694185819141681, −9.888856337544365274733296523082, −7.914288526185822842965897466506, −6.88963940530400534243070420158, −6.26983688690651176611317233748, −5.54088949225152219625628433636, −3.86497570982076923883437131397, −1.72241772577575766359200798985, 1.45309080966147634200016575249, 3.53475827217070640378090679737, 4.63481580558875760451261952540, 5.76125057119510487614907325521, 6.94763319415332191470885424026, 8.189337497121300562891805006405, 9.759733019740318820787061012618, 10.06909888195237766393088343815, 11.70110814862795353636319778708, 11.97003042779243011397372808171

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