Dirichlet series
| L(s) = 1 | − 3.51e13·4-s − 1.58e19·7-s + 1.85e24·13-s + 1.23e27·16-s + 1.05e29·19-s − 2.84e31·25-s + 5.55e32·28-s + 5.28e33·31-s + 2.05e35·37-s − 1.07e37·43-s + 1.42e38·49-s − 6.52e37·52-s − 7.56e39·61-s − 4.35e40·64-s + 2.40e41·67-s − 2.40e41·73-s − 3.72e42·76-s + 7.05e42·79-s − 2.93e43·91-s − 8.57e44·97-s + 9.99e44·100-s + 5.28e44·103-s − 1.28e46·109-s − 1.95e46·112-s + ⋯ |
| L(s) = 1 | − 4-s − 1.52·7-s + 0.160·13-s + 16-s + 1.79·19-s − 25-s + 1.52·28-s + 1.47·31-s + 1.06·37-s − 1.89·43-s + 1.33·49-s − 0.160·52-s − 0.511·61-s − 64-s + 1.97·67-s − 0.285·73-s − 1.79·76-s + 1.41·79-s − 0.244·91-s − 1.70·97-s + 100-s + 0.271·103-s − 1.84·109-s − 1.52·112-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(46-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+45/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(9\) = \(3^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(115.430\) |
| Root analytic conductor: | \(10.7438\) |
| Motivic weight: | \(45\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 9,\ (\ :45/2),\ -1)\) |
Particular Values
| \(L(23)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{47}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + p^{45} T^{2} \) |
| 5 | \( 1 + p^{45} T^{2} \) | |
| 7 | \( 1 + 15801401816335122100 T + p^{45} T^{2} \) | |
| 11 | \( 1 + p^{45} T^{2} \) | |
| 13 | \( 1 - \)\(18\!\cdots\!50\)\( T + p^{45} T^{2} \) | |
| 17 | \( 1 + p^{45} T^{2} \) | |
| 19 | \( 1 - \)\(10\!\cdots\!64\)\( T + p^{45} T^{2} \) | |
| 23 | \( 1 + p^{45} T^{2} \) | |
| 29 | \( 1 + p^{45} T^{2} \) | |
| 31 | \( 1 - \)\(52\!\cdots\!48\)\( T + p^{45} T^{2} \) | |
| 37 | \( 1 - \)\(20\!\cdots\!50\)\( T + p^{45} T^{2} \) | |
| 41 | \( 1 + p^{45} T^{2} \) | |
| 43 | \( 1 + \)\(10\!\cdots\!00\)\( T + p^{45} T^{2} \) | |
| 47 | \( 1 + p^{45} T^{2} \) | |
| 53 | \( 1 + p^{45} T^{2} \) | |
| 59 | \( 1 + p^{45} T^{2} \) | |
| 61 | \( 1 + \)\(75\!\cdots\!98\)\( T + p^{45} T^{2} \) | |
| 67 | \( 1 - \)\(24\!\cdots\!00\)\( T + p^{45} T^{2} \) | |
| 71 | \( 1 + p^{45} T^{2} \) | |
| 73 | \( 1 + \)\(24\!\cdots\!50\)\( T + p^{45} T^{2} \) | |
| 79 | \( 1 - \)\(70\!\cdots\!36\)\( T + p^{45} T^{2} \) | |
| 83 | \( 1 + p^{45} T^{2} \) | |
| 89 | \( 1 + p^{45} T^{2} \) | |
| 97 | \( 1 + \)\(85\!\cdots\!50\)\( T + p^{45} T^{2} \) | |
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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
−11.90972572518301025654500770211, −9.975569010319597559551972732763, −9.422922787982113010659671545655, −7.976740180043810250468979863689, −6.46858675100281636918844718392, −5.27317705126242264570033609778, −3.85073311207011863503861869241, −2.94734635936076071371490258466, −1.01618090874371466944848156383, 0, 1.01618090874371466944848156383, 2.94734635936076071371490258466, 3.85073311207011863503861869241, 5.27317705126242264570033609778, 6.46858675100281636918844718392, 7.976740180043810250468979863689, 9.422922787982113010659671545655, 9.975569010319597559551972732763, 11.90972572518301025654500770211