Properties

Label 2-285-1.1-c9-0-60
Degree $2$
Conductor $285$
Sign $-1$
Analytic cond. $146.785$
Root an. cond. $12.1154$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s + 81·3-s − 476·4-s − 625·5-s − 486·6-s − 5.86e3·7-s + 5.92e3·8-s + 6.56e3·9-s + 3.75e3·10-s − 1.79e4·11-s − 3.85e4·12-s − 6.17e4·13-s + 3.51e4·14-s − 5.06e4·15-s + 2.08e5·16-s + 2.90e5·17-s − 3.93e4·18-s + 1.30e5·19-s + 2.97e5·20-s − 4.75e5·21-s + 1.07e5·22-s − 6.70e5·23-s + 4.80e5·24-s + 3.90e5·25-s + 3.70e5·26-s + 5.31e5·27-s + 2.79e6·28-s + ⋯
L(s)  = 1  − 0.265·2-s + 0.577·3-s − 0.929·4-s − 0.447·5-s − 0.153·6-s − 0.923·7-s + 0.511·8-s + 1/3·9-s + 0.118·10-s − 0.370·11-s − 0.536·12-s − 0.599·13-s + 0.244·14-s − 0.258·15-s + 0.794·16-s + 0.842·17-s − 0.0883·18-s + 0.229·19-s + 0.415·20-s − 0.533·21-s + 0.0981·22-s − 0.499·23-s + 0.295·24-s + 1/5·25-s + 0.158·26-s + 0.192·27-s + 0.858·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(146.785\)
Root analytic conductor: \(12.1154\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 285,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - p^{4} T \)
5 \( 1 + p^{4} T \)
19 \( 1 - p^{4} T \)
good2 \( 1 + 3 p T + p^{9} T^{2} \)
7 \( 1 + 838 p T + p^{9} T^{2} \)
11 \( 1 + 17982 T + p^{9} T^{2} \)
13 \( 1 + 61720 T + p^{9} T^{2} \)
17 \( 1 - 290010 T + p^{9} T^{2} \)
23 \( 1 + 670584 T + p^{9} T^{2} \)
29 \( 1 - 2418540 T + p^{9} T^{2} \)
31 \( 1 - 1323344 T + p^{9} T^{2} \)
37 \( 1 - 10239068 T + p^{9} T^{2} \)
41 \( 1 - 605976 p T + p^{9} T^{2} \)
43 \( 1 + 22280938 T + p^{9} T^{2} \)
47 \( 1 + 9039288 T + p^{9} T^{2} \)
53 \( 1 - 95165394 T + p^{9} T^{2} \)
59 \( 1 - 88064508 T + p^{9} T^{2} \)
61 \( 1 - 159150530 T + p^{9} T^{2} \)
67 \( 1 + 165219640 T + p^{9} T^{2} \)
71 \( 1 + 271185360 T + p^{9} T^{2} \)
73 \( 1 + 167588674 T + p^{9} T^{2} \)
79 \( 1 + 261759544 T + p^{9} T^{2} \)
83 \( 1 + 320534784 T + p^{9} T^{2} \)
89 \( 1 - 443005080 T + p^{9} T^{2} \)
97 \( 1 + 1246029172 T + p^{9} T^{2} \)
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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

−9.845747084348110970387752377462, −8.823898286004877931912657100103, −7.990049143168599806296214341273, −7.17258848431050227314847173182, −5.76127997010599942054663090809, −4.57617304513445175410675923402, −3.62210663165791698722932193844, −2.65678459288515504187963588525, −1.00447179214029417764942625092, 0, 1.00447179214029417764942625092, 2.65678459288515504187963588525, 3.62210663165791698722932193844, 4.57617304513445175410675923402, 5.76127997010599942054663090809, 7.17258848431050227314847173182, 7.990049143168599806296214341273, 8.823898286004877931912657100103, 9.845747084348110970387752377462

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