Dirichlet series
| L(s) = 1 | + 5·7-s + 7·13-s + 19-s − 5·25-s − 4·31-s + 37-s − 8·43-s + 18·49-s + 13·61-s − 11·67-s + 17·73-s − 13·79-s + 35·91-s + 5·97-s − 7·103-s − 2·109-s + ⋯ |
| L(s) = 1 | + 1.88·7-s + 1.94·13-s + 0.229·19-s − 25-s − 0.718·31-s + 0.164·37-s − 1.21·43-s + 18/7·49-s + 1.66·61-s − 1.34·67-s + 1.98·73-s − 1.46·79-s + 3.66·91-s + 0.507·97-s − 0.689·103-s − 0.191·109-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(1728\) = \(2^{6} \cdot 3^{3}\) |
| Sign: | $1$ |
| Analytic conductor: | \(13.7981\) |
| Root analytic conductor: | \(3.71458\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((2,\ 1728,\ (\ :1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.360844297\) |
| \(L(\frac12)\) | \(\approx\) | \(2.360844297\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ | |
|---|---|---|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | ||
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af | |
| 11 | \( 1 + p T^{2} \) | 1.11.a | |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah | |
| 17 | \( 1 + p T^{2} \) | 1.17.a | |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab | |
| 23 | \( 1 + p T^{2} \) | 1.23.a | |
| 29 | \( 1 + p T^{2} \) | 1.29.a | |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e | |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab | |
| 41 | \( 1 + p T^{2} \) | 1.41.a | |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i | |
| 47 | \( 1 + p T^{2} \) | 1.47.a | |
| 53 | \( 1 + p T^{2} \) | 1.53.a | |
| 59 | \( 1 + p T^{2} \) | 1.59.a | |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an | |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l | |
| 71 | \( 1 + p T^{2} \) | 1.71.a | |
| 73 | \( 1 - 17 T + p T^{2} \) | 1.73.ar | |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n | |
| 83 | \( 1 + p T^{2} \) | 1.83.a | |
| 89 | \( 1 + p T^{2} \) | 1.89.a | |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af | |
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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
−9.125589838381877994041862400986, −8.360413074571016534853094652479, −8.001529503824122494641395863146, −7.02193571198558069600312241264, −5.94215330426161837619813679812, −5.28577670060288457890184502723, −4.31915400924746551561976811972, −3.53995560336243251785891738493, −2.01230440740899716446600775646, −1.20941721581339133701993176268, 1.20941721581339133701993176268, 2.01230440740899716446600775646, 3.53995560336243251785891738493, 4.31915400924746551561976811972, 5.28577670060288457890184502723, 5.94215330426161837619813679812, 7.02193571198558069600312241264, 8.001529503824122494641395863146, 8.360413074571016534853094652479, 9.125589838381877994041862400986