Properties

Label 2-15e2-225.121-c1-0-4
Degree $2$
Conductor $225$
Sign $-0.906 - 0.422i$
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  + (−1.39 + 0.297i)2-s + (−1.19 + 1.25i)3-s + (0.0411 − 0.0183i)4-s + (1.76 + 1.37i)5-s + (1.30 − 2.10i)6-s + (1.92 + 3.32i)7-s + (2.26 − 1.64i)8-s + (−0.129 − 2.99i)9-s + (−2.87 − 1.39i)10-s + (−5.11 + 1.08i)11-s + (−0.0263 + 0.0733i)12-s + (3.28 + 0.699i)13-s + (−3.67 − 4.08i)14-s + (−3.83 + 0.560i)15-s + (−2.73 + 3.03i)16-s + (−3.90 + 2.84i)17-s + ⋯
L(s)  = 1  + (−0.989 + 0.210i)2-s + (−0.691 + 0.722i)3-s + (0.0205 − 0.00915i)4-s + (0.788 + 0.614i)5-s + (0.532 − 0.859i)6-s + (0.725 + 1.25i)7-s + (0.799 − 0.580i)8-s + (−0.0431 − 0.999i)9-s + (−0.909 − 0.442i)10-s + (−1.54 + 0.327i)11-s + (−0.00760 + 0.0211i)12-s + (0.912 + 0.193i)13-s + (−0.982 − 1.09i)14-s + (−0.989 + 0.144i)15-s + (−0.683 + 0.759i)16-s + (−0.948 + 0.688i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.906 - 0.422i$
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: \((2,\ 225,\ (\ :1/2),\ -0.906 - 0.422i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.112941 + 0.510053i\)
\(L(\frac12)\) \(\approx\) \(0.112941 + 0.510053i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.19 - 1.25i)T \)
5 \( 1 + (-1.76 - 1.37i)T \)
good2 \( 1 + (1.39 - 0.297i)T + (1.82 - 0.813i)T^{2} \)
7 \( 1 + (-1.92 - 3.32i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.11 - 1.08i)T + (10.0 - 4.47i)T^{2} \)
13 \( 1 + (-3.28 - 0.699i)T + (11.8 + 5.28i)T^{2} \)
17 \( 1 + (3.90 - 2.84i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.90 - 1.38i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (3.12 + 3.46i)T + (-2.40 + 22.8i)T^{2} \)
29 \( 1 + (-0.768 + 7.30i)T + (-28.3 - 6.02i)T^{2} \)
31 \( 1 + (0.0679 + 0.646i)T + (-30.3 + 6.44i)T^{2} \)
37 \( 1 + (-0.315 - 0.970i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.66 - 0.779i)T + (37.4 + 16.6i)T^{2} \)
43 \( 1 + (2.34 + 4.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.24 - 11.8i)T + (-45.9 - 9.77i)T^{2} \)
53 \( 1 + (-2.92 - 2.12i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-7.61 - 1.61i)T + (53.8 + 23.9i)T^{2} \)
61 \( 1 + (-6.44 + 1.36i)T + (55.7 - 24.8i)T^{2} \)
67 \( 1 + (-0.106 - 1.01i)T + (-65.5 + 13.9i)T^{2} \)
71 \( 1 + (2.82 + 2.04i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.38 - 4.25i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.171 - 1.63i)T + (-77.2 - 16.4i)T^{2} \)
83 \( 1 + (-10.3 - 4.59i)T + (55.5 + 61.6i)T^{2} \)
89 \( 1 + (0.935 - 2.87i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (1.69 - 16.1i)T + (-94.8 - 20.1i)T^{2} \)
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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

−12.57494973433493222953359463926, −11.21951271273249625628543350844, −10.56584784787241131258471683249, −9.814000736436978934981547350543, −8.803664245696690895828708532076, −8.052543008159287841590109876960, −6.44597184261749273084714710654, −5.59057446422509405049481098539, −4.34718800878149595777024269790, −2.24247818473773957300388158690, 0.65958515409849120724929439821, 1.94552412664851846094799241911, 4.70792200155913329225811598654, 5.49939728095603011526236159599, 7.02285926513973464319514421285, 8.010794451067614702359593838486, 8.722565226831416797884340388041, 10.17506641008297948224112810401, 10.70389363727504070430695030969, 11.41700747571274211083860160739

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