Dirichlet series
| L(s) = 1 | − 5.49e11·4-s + 5.45e16·7-s + 1.05e22·13-s + 3.02e23·16-s − 1.67e25·19-s − 1.81e27·25-s − 3.00e28·28-s + 2.36e29·31-s − 5.15e29·37-s − 6.23e31·43-s + 2.07e33·49-s − 5.79e33·52-s + 7.81e34·61-s − 1.66e35·64-s + 3.61e35·67-s − 2.95e36·73-s + 9.23e36·76-s + 1.09e37·79-s + 5.75e38·91-s + 6.09e38·97-s + 9.99e38·100-s + 3.27e39·103-s + 6.05e39·109-s + 1.65e40·112-s + ⋯ |
| L(s) = 1 | − 4-s + 1.81·7-s + 1.99·13-s + 16-s − 1.94·19-s − 25-s − 1.81·28-s + 1.96·31-s − 0.135·37-s − 0.875·43-s + 2.27·49-s − 1.99·52-s + 1.19·61-s − 64-s + 0.890·67-s − 1.36·73-s + 1.94·76-s + 1.08·79-s + 3.61·91-s + 1.10·97-s + 100-s + 1.83·103-s + 1.12·109-s + 1.81·112-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+39/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(9\) = \(3^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(86.7055\) |
| Root analytic conductor: | \(9.31158\) |
| Motivic weight: | \(39\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((2,\ 9,\ (\ :39/2),\ 1)\) |
Particular Values
| \(L(20)\) | \(\approx\) | \(2.295304899\) |
| \(L(\frac12)\) | \(\approx\) | \(2.295304899\) |
| \(L(\frac{41}{2})\) | 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 | \( 1 + p^{39} T^{2} \) |
| 5 | \( 1 + p^{39} T^{2} \) | |
| 7 | \( 1 - 54595696320612740 T + p^{39} T^{2} \) | |
| 11 | \( 1 + p^{39} T^{2} \) | |
| 13 | \( 1 - \)\(10\!\cdots\!10\)\( T + p^{39} T^{2} \) | |
| 17 | \( 1 + p^{39} T^{2} \) | |
| 19 | \( 1 + \)\(16\!\cdots\!04\)\( T + p^{39} T^{2} \) | |
| 23 | \( 1 + p^{39} T^{2} \) | |
| 29 | \( 1 + p^{39} T^{2} \) | |
| 31 | \( 1 - \)\(23\!\cdots\!28\)\( T + p^{39} T^{2} \) | |
| 37 | \( 1 + \)\(51\!\cdots\!30\)\( T + p^{39} T^{2} \) | |
| 41 | \( 1 + p^{39} T^{2} \) | |
| 43 | \( 1 + \)\(62\!\cdots\!40\)\( T + p^{39} T^{2} \) | |
| 47 | \( 1 + p^{39} T^{2} \) | |
| 53 | \( 1 + p^{39} T^{2} \) | |
| 59 | \( 1 + p^{39} T^{2} \) | |
| 61 | \( 1 - \)\(78\!\cdots\!42\)\( T + p^{39} T^{2} \) | |
| 67 | \( 1 - \)\(36\!\cdots\!40\)\( T + p^{39} T^{2} \) | |
| 71 | \( 1 + p^{39} T^{2} \) | |
| 73 | \( 1 + \)\(29\!\cdots\!70\)\( T + p^{39} T^{2} \) | |
| 79 | \( 1 - \)\(10\!\cdots\!24\)\( T + p^{39} T^{2} \) | |
| 83 | \( 1 + p^{39} T^{2} \) | |
| 89 | \( 1 + p^{39} T^{2} \) | |
| 97 | \( 1 - \)\(60\!\cdots\!90\)\( T + p^{39} T^{2} \) | |
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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
−13.23947576196898702573942675573, −11.56520710851433341329147626018, −10.45333089608224535887390603575, −8.586595117442634030596707989710, −8.199415624936887350475767345013, −6.07585039217707021159473315997, −4.72784031627459995615242738032, −3.88422720816525319940841617288, −1.84954485814519081496042544288, −0.813823125213642985870557174692, 0.813823125213642985870557174692, 1.84954485814519081496042544288, 3.88422720816525319940841617288, 4.72784031627459995615242738032, 6.07585039217707021159473315997, 8.199415624936887350475767345013, 8.586595117442634030596707989710, 10.45333089608224535887390603575, 11.56520710851433341329147626018, 13.23947576196898702573942675573