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Properties

Label 2-15e2-9.4-c1-0-3
Degree $2$
Conductor $225$
Sign $0.230 - 0.973i$
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  + (0.736 + 1.27i)2-s + (−1.69 − 0.350i)3-s + (−0.0852 + 0.147i)4-s + (−0.802 − 2.42i)6-s + (1.93 + 3.34i)7-s + 2.69·8-s + (2.75 + 1.18i)9-s + (−0.130 − 0.225i)11-s + (0.196 − 0.220i)12-s + (−2.03 + 3.53i)13-s + (−2.84 + 4.93i)14-s + (2.15 + 3.73i)16-s − 3.26·17-s + (0.513 + 4.38i)18-s + 4.24·19-s + ⋯
L(s)  = 1  + (0.520 + 0.902i)2-s + (−0.979 − 0.202i)3-s + (−0.0426 + 0.0738i)4-s + (−0.327 − 0.988i)6-s + (0.730 + 1.26i)7-s + 0.952·8-s + (0.918 + 0.395i)9-s + (−0.0392 − 0.0679i)11-s + (0.0566 − 0.0636i)12-s + (−0.565 + 0.979i)13-s + (−0.761 + 1.31i)14-s + (0.538 + 0.933i)16-s − 0.790·17-s + (0.121 + 1.03i)18-s + 0.974·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.230 - 0.973i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.230 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07016 + 0.846273i\)
\(L(\frac12)\) \(\approx\) \(1.07016 + 0.846273i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−12.40070569077270433555718522092, −11.56945779305118771394788176270, −10.83198255200568121939487556847, −9.492409780164119104221604243941, −8.222283500620870492467705295803, −7.02162700323063525793107221512, −6.28385813597708571161919513401, −5.18776344705136876850609528611, −4.65081063695935285641788332471, −1.98376703373363187207888919375, 1.31745649835897943992736878638, 3.37838439955640104859898617918, 4.53323122573373513863074045733, 5.33899724752617185621038524129, 7.12883054827753877302501432679, 7.68868247394905402439514524421, 9.647238709757498182648054049213, 10.55819202114526556531236464670, 11.17834295968714688292547032797, 11.79310569764523210445101883671

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