Properties

Label 2-15e2-1.1-c3-0-21
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.35·2-s + 3.28·4-s − 30.4·7-s − 15.8·8-s − 31.4·11-s + 60.7·13-s − 102.·14-s − 79.4·16-s − 121.·17-s − 14.4·19-s − 105.·22-s + 13.6·23-s + 204.·26-s − 99.8·28-s + 76.0·29-s + 183.·31-s − 140.·32-s − 407.·34-s + 37.3·37-s − 48.4·38-s + 30.6·41-s − 327.·43-s − 103.·44-s + 45.9·46-s − 449.·47-s + 583.·49-s + 199.·52-s + ⋯
L(s)  = 1  + 1.18·2-s + 0.410·4-s − 1.64·7-s − 0.700·8-s − 0.861·11-s + 1.29·13-s − 1.95·14-s − 1.24·16-s − 1.72·17-s − 0.174·19-s − 1.02·22-s + 0.124·23-s + 1.53·26-s − 0.674·28-s + 0.486·29-s + 1.06·31-s − 0.774·32-s − 2.05·34-s + 0.166·37-s − 0.206·38-s + 0.116·41-s − 1.16·43-s − 0.353·44-s + 0.147·46-s − 1.39·47-s + 1.70·49-s + 0.531·52-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: \(1\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.35T + 8T^{2} \)
7 \( 1 + 30.4T + 343T^{2} \)
11 \( 1 + 31.4T + 1.33e3T^{2} \)
13 \( 1 - 60.7T + 2.19e3T^{2} \)
17 \( 1 + 121.T + 4.91e3T^{2} \)
19 \( 1 + 14.4T + 6.85e3T^{2} \)
23 \( 1 - 13.6T + 1.21e4T^{2} \)
29 \( 1 - 76.0T + 2.43e4T^{2} \)
31 \( 1 - 183.T + 2.97e4T^{2} \)
37 \( 1 - 37.3T + 5.06e4T^{2} \)
41 \( 1 - 30.6T + 6.89e4T^{2} \)
43 \( 1 + 327.T + 7.95e4T^{2} \)
47 \( 1 + 449.T + 1.03e5T^{2} \)
53 \( 1 + 301.T + 1.48e5T^{2} \)
59 \( 1 + 340.T + 2.05e5T^{2} \)
61 \( 1 - 619.T + 2.26e5T^{2} \)
67 \( 1 + 256.T + 3.00e5T^{2} \)
71 \( 1 + 499.T + 3.57e5T^{2} \)
73 \( 1 - 19.1T + 3.89e5T^{2} \)
79 \( 1 - 257.T + 4.93e5T^{2} \)
83 \( 1 - 914.T + 5.71e5T^{2} \)
89 \( 1 - 1.05e3T + 7.04e5T^{2} \)
97 \( 1 + 521T + 9.12e5T^{2} \)
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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

−11.55021960122252411399523205809, −10.48855040014758050701933772578, −9.365353954956778813155799824565, −8.428982533696333969387209869995, −6.60261573530341376257426617060, −6.21528123781056840485155092775, −4.83743015699844166885344310706, −3.67197270933323698391309349481, −2.71263975134744512745668581964, 0, 2.71263975134744512745668581964, 3.67197270933323698391309349481, 4.83743015699844166885344310706, 6.21528123781056840485155092775, 6.60261573530341376257426617060, 8.428982533696333969387209869995, 9.365353954956778813155799824565, 10.48855040014758050701933772578, 11.55021960122252411399523205809

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