Properties

Label 2-15e2-1.1-c11-0-52
Degree $2$
Conductor $225$
Sign $-1$
Analytic cond. $172.877$
Root an. cond. $13.1482$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 24·2-s − 1.47e3·4-s + 1.67e4·7-s + 8.44e4·8-s − 5.34e5·11-s + 5.77e5·13-s − 4.01e5·14-s + 9.87e5·16-s − 6.90e6·17-s + 1.06e7·19-s + 1.28e7·22-s + 1.86e7·23-s − 1.38e7·26-s − 2.46e7·28-s − 1.28e8·29-s − 5.28e7·31-s − 1.96e8·32-s + 1.65e8·34-s + 1.82e8·37-s − 2.55e8·38-s − 3.08e8·41-s + 1.71e7·43-s + 7.86e8·44-s − 4.47e8·46-s + 2.68e9·47-s − 1.69e9·49-s − 8.50e8·52-s + ⋯
L(s)  = 1  − 0.530·2-s − 0.718·4-s + 0.376·7-s + 0.911·8-s − 1.00·11-s + 0.431·13-s − 0.199·14-s + 0.235·16-s − 1.17·17-s + 0.987·19-s + 0.530·22-s + 0.603·23-s − 0.228·26-s − 0.270·28-s − 1.16·29-s − 0.331·31-s − 1.03·32-s + 0.625·34-s + 0.431·37-s − 0.523·38-s − 0.415·41-s + 0.0177·43-s + 0.719·44-s − 0.320·46-s + 1.70·47-s − 0.858·49-s − 0.310·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(172.877\)
Root analytic conductor: \(13.1482\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 225,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) 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 \( 1 + 3 p^{3} T + p^{11} T^{2} \)
7 \( 1 - 2392 p T + p^{11} T^{2} \)
11 \( 1 + 534612 T + p^{11} T^{2} \)
13 \( 1 - 577738 T + p^{11} T^{2} \)
17 \( 1 + 6905934 T + p^{11} T^{2} \)
19 \( 1 - 10661420 T + p^{11} T^{2} \)
23 \( 1 - 18643272 T + p^{11} T^{2} \)
29 \( 1 + 128406630 T + p^{11} T^{2} \)
31 \( 1 + 52843168 T + p^{11} T^{2} \)
37 \( 1 - 182213314 T + p^{11} T^{2} \)
41 \( 1 + 308120442 T + p^{11} T^{2} \)
43 \( 1 - 17125708 T + p^{11} T^{2} \)
47 \( 1 - 2687348496 T + p^{11} T^{2} \)
53 \( 1 + 1596055698 T + p^{11} T^{2} \)
59 \( 1 - 5189203740 T + p^{11} T^{2} \)
61 \( 1 - 6956478662 T + p^{11} T^{2} \)
67 \( 1 - 15481826884 T + p^{11} T^{2} \)
71 \( 1 + 9791485272 T + p^{11} T^{2} \)
73 \( 1 + 1463791322 T + p^{11} T^{2} \)
79 \( 1 - 38116845680 T + p^{11} T^{2} \)
83 \( 1 + 29335099668 T + p^{11} T^{2} \)
89 \( 1 - 24992917110 T + p^{11} T^{2} \)
97 \( 1 + 75013568546 T + p^{11} T^{2} \)
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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

−9.662572762095747143132780674942, −8.843852527746543530606281541160, −7.991682645195826562615795853972, −7.11934858179432313599510331027, −5.57944796469263924601830309439, −4.77376602138295544567223008596, −3.64430619700881191000754583992, −2.22421868028396419396015970050, −1.00955341940831894074426466763, 0, 1.00955341940831894074426466763, 2.22421868028396419396015970050, 3.64430619700881191000754583992, 4.77376602138295544567223008596, 5.57944796469263924601830309439, 7.11934858179432313599510331027, 7.991682645195826562615795853972, 8.843852527746543530606281541160, 9.662572762095747143132780674942

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