Properties

Label 2-15e2-1.1-c3-0-0
Degree $2$
Conductor $225$
Sign $1$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.70·2-s − 5.10·4-s − 22.2·7-s + 22.2·8-s + 1.79·11-s − 58.2·13-s + 37.7·14-s + 2.89·16-s + 18.9·17-s + 104.·19-s − 3.04·22-s − 49.6·23-s + 99.0·26-s + 113.·28-s + 293.·29-s + 64.4·31-s − 183.·32-s − 32.3·34-s + 19.8·37-s − 178.·38-s + 165.·41-s + 247.·43-s − 9.14·44-s + 84.4·46-s + 384.·47-s + 150.·49-s + 297.·52-s + ⋯
L(s)  = 1  − 0.601·2-s − 0.638·4-s − 1.19·7-s + 0.985·8-s + 0.0490·11-s − 1.24·13-s + 0.721·14-s + 0.0452·16-s + 0.270·17-s + 1.26·19-s − 0.0295·22-s − 0.449·23-s + 0.747·26-s + 0.765·28-s + 1.87·29-s + 0.373·31-s − 1.01·32-s − 0.162·34-s + 0.0883·37-s − 0.761·38-s + 0.630·41-s + 0.877·43-s − 0.0313·44-s + 0.270·46-s + 1.19·47-s + 0.438·49-s + 0.792·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7797505016\)
\(L(\frac12)\) \(\approx\) \(0.7797505016\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + 1.70T + 8T^{2} \)
7 \( 1 + 22.2T + 343T^{2} \)
11 \( 1 - 1.79T + 1.33e3T^{2} \)
13 \( 1 + 58.2T + 2.19e3T^{2} \)
17 \( 1 - 18.9T + 4.91e3T^{2} \)
19 \( 1 - 104.T + 6.85e3T^{2} \)
23 \( 1 + 49.6T + 1.21e4T^{2} \)
29 \( 1 - 293.T + 2.43e4T^{2} \)
31 \( 1 - 64.4T + 2.97e4T^{2} \)
37 \( 1 - 19.8T + 5.06e4T^{2} \)
41 \( 1 - 165.T + 6.89e4T^{2} \)
43 \( 1 - 247.T + 7.95e4T^{2} \)
47 \( 1 - 384.T + 1.03e5T^{2} \)
53 \( 1 - 463.T + 1.48e5T^{2} \)
59 \( 1 - 73.7T + 2.05e5T^{2} \)
61 \( 1 + 137.T + 2.26e5T^{2} \)
67 \( 1 + 173.T + 3.00e5T^{2} \)
71 \( 1 - 594.T + 3.57e5T^{2} \)
73 \( 1 + 320.T + 3.89e5T^{2} \)
79 \( 1 + 770.T + 4.93e5T^{2} \)
83 \( 1 - 173.T + 5.71e5T^{2} \)
89 \( 1 + 1.01e3T + 7.04e5T^{2} \)
97 \( 1 + 384.T + 9.12e5T^{2} \)
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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

−11.94903704569756358426326497767, −10.36240979570695940946407032855, −9.809125067411754901408620420159, −9.074810765468157147871308543604, −7.87282757374301840820356457019, −6.94352074344351613310366883575, −5.55077672619013929452988806514, −4.29877697518416051588241199469, −2.85016827522063193080890320155, −0.71535180307929917258243212967, 0.71535180307929917258243212967, 2.85016827522063193080890320155, 4.29877697518416051588241199469, 5.55077672619013929452988806514, 6.94352074344351613310366883575, 7.87282757374301840820356457019, 9.074810765468157147871308543604, 9.809125067411754901408620420159, 10.36240979570695940946407032855, 11.94903704569756358426326497767

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