Properties

Label 2-15e2-225.121-c1-0-17
Degree $2$
Conductor $225$
Sign $0.989 - 0.146i$
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  + (1.37 − 0.292i)2-s + (1.63 + 0.572i)3-s + (−0.0128 + 0.00572i)4-s + (1.56 + 1.59i)5-s + (2.42 + 0.310i)6-s + (−2.48 − 4.30i)7-s + (−2.29 + 1.66i)8-s + (2.34 + 1.87i)9-s + (2.62 + 1.74i)10-s + (3.07 − 0.653i)11-s + (−0.0242 + 0.00199i)12-s + (−1.44 − 0.307i)13-s + (−4.68 − 5.20i)14-s + (1.63 + 3.50i)15-s + (−2.65 + 2.95i)16-s + (−2.39 + 1.73i)17-s + ⋯
L(s)  = 1  + (0.974 − 0.207i)2-s + (0.943 + 0.330i)3-s + (−0.00642 + 0.00286i)4-s + (0.698 + 0.715i)5-s + (0.988 + 0.126i)6-s + (−0.939 − 1.62i)7-s + (−0.811 + 0.589i)8-s + (0.781 + 0.624i)9-s + (0.829 + 0.552i)10-s + (0.926 − 0.196i)11-s + (−0.00701 + 0.000574i)12-s + (−0.401 − 0.0852i)13-s + (−1.25 − 1.39i)14-s + (0.422 + 0.906i)15-s + (−0.664 + 0.737i)16-s + (−0.579 + 0.421i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.989 - 0.146i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.989 - 0.146i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31760 + 0.170640i\)
\(L(\frac12)\) \(\approx\) \(2.31760 + 0.170640i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.63 - 0.572i)T \)
5 \( 1 + (-1.56 - 1.59i)T \)
good2 \( 1 + (-1.37 + 0.292i)T + (1.82 - 0.813i)T^{2} \)
7 \( 1 + (2.48 + 4.30i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.07 + 0.653i)T + (10.0 - 4.47i)T^{2} \)
13 \( 1 + (1.44 + 0.307i)T + (11.8 + 5.28i)T^{2} \)
17 \( 1 + (2.39 - 1.73i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.51 - 1.09i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (4.45 + 4.94i)T + (-2.40 + 22.8i)T^{2} \)
29 \( 1 + (-0.427 + 4.06i)T + (-28.3 - 6.02i)T^{2} \)
31 \( 1 + (0.681 + 6.47i)T + (-30.3 + 6.44i)T^{2} \)
37 \( 1 + (-1.50 - 4.62i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.38 - 0.508i)T + (37.4 + 16.6i)T^{2} \)
43 \( 1 + (1.92 + 3.32i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.175 - 1.66i)T + (-45.9 - 9.77i)T^{2} \)
53 \( 1 + (-0.228 - 0.165i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.95 - 0.628i)T + (53.8 + 23.9i)T^{2} \)
61 \( 1 + (-7.58 + 1.61i)T + (55.7 - 24.8i)T^{2} \)
67 \( 1 + (-1.19 - 11.3i)T + (-65.5 + 13.9i)T^{2} \)
71 \( 1 + (-13.0 - 9.48i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.03 - 6.27i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.847 - 8.05i)T + (-77.2 - 16.4i)T^{2} \)
83 \( 1 + (8.79 + 3.91i)T + (55.5 + 61.6i)T^{2} \)
89 \( 1 + (-0.845 + 2.60i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-1.02 + 9.77i)T + (-94.8 - 20.1i)T^{2} \)
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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

−12.82790580454708629356058458437, −11.28439934632494687537128157693, −10.15218572836180032051909179437, −9.659097339988412640339219275100, −8.368914382927969009843289822972, −7.00524019539395298404715897090, −6.13780753411361562465083194430, −4.26245835247463382675933003873, −3.76457528462634113470042137279, −2.50387077765487573436184945049, 2.17961953511117110306359509815, 3.48415852635062427733166558886, 4.88661984508902848755460951733, 5.98595082073876292891746879050, 6.77770299914015207163013174293, 8.628069918708017237615859416400, 9.279139818695227121977177456975, 9.677764422542375919337614092166, 12.01374007327691190620175103561, 12.47722561456311461542366442072

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