Properties

Label 2-15e2-1.1-c5-0-15
Degree $2$
Conductor $225$
Sign $1$
Analytic cond. $36.0863$
Root an. cond. $6.00719$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·2-s + 17·4-s − 12·7-s − 105·8-s − 112·11-s + 974·13-s − 84·14-s − 1.27e3·16-s + 2.18e3·17-s + 1.42e3·19-s − 784·22-s + 3.21e3·23-s + 6.81e3·26-s − 204·28-s + 4.15e3·29-s − 5.68e3·31-s − 5.59e3·32-s + 1.52e4·34-s − 6.48e3·37-s + 9.94e3·38-s − 5.40e3·41-s + 2.17e4·43-s − 1.90e3·44-s + 2.25e4·46-s − 368·47-s − 1.66e4·49-s + 1.65e4·52-s + ⋯
L(s)  = 1  + 1.23·2-s + 0.531·4-s − 0.0925·7-s − 0.580·8-s − 0.279·11-s + 1.59·13-s − 0.114·14-s − 1.24·16-s + 1.83·17-s + 0.902·19-s − 0.345·22-s + 1.26·23-s + 1.97·26-s − 0.0491·28-s + 0.916·29-s − 1.06·31-s − 0.965·32-s + 2.26·34-s − 0.778·37-s + 1.11·38-s − 0.501·41-s + 1.79·43-s − 0.148·44-s + 1.56·46-s − 0.0242·47-s − 0.991·49-s + 0.849·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(36.0863\)
Root analytic conductor: \(6.00719\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.930095076\)
\(L(\frac12)\) \(\approx\) \(3.930095076\)
\(L(\frac{7}{2})\) 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 \( 1 - 7 T + p^{5} T^{2} \)
7 \( 1 + 12 T + p^{5} T^{2} \)
11 \( 1 + 112 T + p^{5} T^{2} \)
13 \( 1 - 974 T + p^{5} T^{2} \)
17 \( 1 - 2182 T + p^{5} T^{2} \)
19 \( 1 - 1420 T + p^{5} T^{2} \)
23 \( 1 - 3216 T + p^{5} T^{2} \)
29 \( 1 - 4150 T + p^{5} T^{2} \)
31 \( 1 + 5688 T + p^{5} T^{2} \)
37 \( 1 + 6482 T + p^{5} T^{2} \)
41 \( 1 + 5402 T + p^{5} T^{2} \)
43 \( 1 - 21764 T + p^{5} T^{2} \)
47 \( 1 + 368 T + p^{5} T^{2} \)
53 \( 1 - 12586 T + p^{5} T^{2} \)
59 \( 1 - 25520 T + p^{5} T^{2} \)
61 \( 1 - 11782 T + p^{5} T^{2} \)
67 \( 1 - 13188 T + p^{5} T^{2} \)
71 \( 1 - 35968 T + p^{5} T^{2} \)
73 \( 1 + 73186 T + p^{5} T^{2} \)
79 \( 1 + 52440 T + p^{5} T^{2} \)
83 \( 1 - 69036 T + p^{5} T^{2} \)
89 \( 1 - 33870 T + p^{5} T^{2} \)
97 \( 1 + 143042 T + p^{5} T^{2} \)
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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

−11.64071821108126587467353403299, −10.66941620185424166240274780871, −9.426170476392902061114971155685, −8.381789924735077784194262862201, −7.07422978957815736952526605020, −5.84874365943698712673201860249, −5.18999256761872855798852254839, −3.77175185121352325568480165215, −3.03326451516435357143866784945, −1.06859929759986495696275945157, 1.06859929759986495696275945157, 3.03326451516435357143866784945, 3.77175185121352325568480165215, 5.18999256761872855798852254839, 5.84874365943698712673201860249, 7.07422978957815736952526605020, 8.381789924735077784194262862201, 9.426170476392902061114971155685, 10.66941620185424166240274780871, 11.64071821108126587467353403299

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