Among degree 2 L-functions of (motivic) weight greater than 1, this is the one of rank 2 with the smallest analytic conductor.
Dirichlet series
| L(s) = 1 | − 2-s − 8·3-s − 7·4-s − 15·5-s + 8·6-s − 25·7-s + 15·8-s + 37·9-s + 15·10-s − 51·11-s + 56·12-s + 2·13-s + 25·14-s + 120·15-s + 41·16-s + 31·17-s − 37·18-s − 123·19-s + 105·20-s + 200·21-s + 51·22-s − 149·23-s − 120·24-s + 100·25-s − 2·26-s − 80·27-s + 175·28-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 1.53·3-s − 7/8·4-s − 1.34·5-s + 0.544·6-s − 1.34·7-s + 0.662·8-s + 1.37·9-s + 0.474·10-s − 1.39·11-s + 1.34·12-s + 0.0426·13-s + 0.477·14-s + 2.06·15-s + 0.640·16-s + 0.442·17-s − 0.484·18-s − 1.48·19-s + 1.17·20-s + 2.07·21-s + 0.494·22-s − 1.35·23-s − 1.02·24-s + 4/5·25-s − 0.0150·26-s − 0.570·27-s + 1.18·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(127\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.49324\) |
| Root analytic conductor: | \(2.73737\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((2,\ 127,\ (\ :3/2),\ 1)\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 127 | \( 1 + p T \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 3 | \( 1 + 8 T + p^{3} T^{2} \) | |
| 5 | \( 1 + 3 p T + p^{3} T^{2} \) | |
| 7 | \( 1 + 25 T + p^{3} T^{2} \) | |
| 11 | \( 1 + 51 T + p^{3} T^{2} \) | |
| 13 | \( 1 - 2 T + p^{3} T^{2} \) | |
| 17 | \( 1 - 31 T + p^{3} T^{2} \) | |
| 19 | \( 1 + 123 T + p^{3} T^{2} \) | |
| 23 | \( 1 + 149 T + p^{3} T^{2} \) | |
| 29 | \( 1 - 6 T + p^{3} T^{2} \) | |
| 31 | \( 1 - 10 T + p^{3} T^{2} \) | |
| 37 | \( 1 + 348 T + p^{3} T^{2} \) | |
| 41 | \( 1 + 387 T + p^{3} T^{2} \) | |
| 43 | \( 1 + 80 T + p^{3} T^{2} \) | |
| 47 | \( 1 - 266 T + p^{3} T^{2} \) | |
| 53 | \( 1 - 347 T + p^{3} T^{2} \) | |
| 59 | \( 1 + 656 T + p^{3} T^{2} \) | |
| 61 | \( 1 + 158 T + p^{3} T^{2} \) | |
| 67 | \( 1 + 314 T + p^{3} T^{2} \) | |
| 71 | \( 1 - 312 T + p^{3} T^{2} \) | |
| 73 | \( 1 + 646 T + p^{3} T^{2} \) | |
| 79 | \( 1 + 846 T + p^{3} T^{2} \) | |
| 83 | \( 1 - 1352 T + p^{3} T^{2} \) | |
| 89 | \( 1 - 1242 T + p^{3} T^{2} \) | |
| 97 | \( 1 - 632 T + p^{3} T^{2} \) | |
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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
−12.02007571296655027007819446571, −10.57571149678877312196382124665, −10.18409261044263319099076414047, −8.556207113811368295034915563048, −7.45774865157933390598502978172, −6.14789823192737351587679669776, −4.92412076437925280300225549780, −3.72975010343758748498352902668, 0, 0, 3.72975010343758748498352902668, 4.92412076437925280300225549780, 6.14789823192737351587679669776, 7.45774865157933390598502978172, 8.556207113811368295034915563048, 10.18409261044263319099076414047, 10.57571149678877312196382124665, 12.02007571296655027007819446571