Error: no document with id 196017213 found in table mf_hecke_traces.
Dirichlet series
| L(s) = 1 | + (−5.04 + 2.91i)5-s + (−5.74 + 9.94i)7-s + (12.6 + 7.32i)11-s + (9.48 + 16.4i)13-s + 5.31i·17-s + 8.97·19-s + (14.1 − 8.14i)23-s + (4.48 − 7.76i)25-s + (36.3 + 21i)29-s + (−4.74 − 8.21i)31-s − 66.9i·35-s − 52.9·37-s + (−54.4 + 31.4i)41-s + (1.48 − 2.57i)43-s + (65.7 + 37.9i)47-s + ⋯ |
| L(s) = 1 | + (−1.00 + 0.582i)5-s + (−0.820 + 1.42i)7-s + (1.15 + 0.666i)11-s + (0.729 + 1.26i)13-s + 0.312i·17-s + 0.472·19-s + (0.613 − 0.354i)23-s + (0.179 − 0.310i)25-s + (1.25 + 0.724i)29-s + (−0.152 − 0.264i)31-s − 1.91i·35-s − 1.43·37-s + (−1.32 + 0.767i)41-s + (0.0345 − 0.0598i)43-s + (1.39 + 0.807i)47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(1296\) = \(2^{4} \cdot 3^{4}\) |
| Sign: | $-0.984 - 0.173i$ |
| Analytic conductor: | \(35.3134\) |
| Root analytic conductor: | \(5.94251\) |
| Motivic weight: | \(2\) |
| Rational: | no |
| Arithmetic: | yes |
| Character: | $\chi_{1296} (593, \cdot )$ |
| Primitive: | yes |
| Self-dual: | no |
| Analytic rank: | \(0\) |
| Selberg data: | \((2,\ 1296,\ (\ :1),\ -0.984 - 0.173i)\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(1.226742380\) |
| \(L(\frac12)\) | \(\approx\) | \(1.226742380\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 + (5.04 - 2.91i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (5.74 - 9.94i)T + (-24.5 - 42.4i)T^{2} \) | |
| 11 | \( 1 + (-12.6 - 7.32i)T + (60.5 + 104. i)T^{2} \) | |
| 13 | \( 1 + (-9.48 - 16.4i)T + (-84.5 + 146. i)T^{2} \) | |
| 17 | \( 1 - 5.31iT - 289T^{2} \) | |
| 19 | \( 1 - 8.97T + 361T^{2} \) | |
| 23 | \( 1 + (-14.1 + 8.14i)T + (264.5 - 458. i)T^{2} \) | |
| 29 | \( 1 + (-36.3 - 21i)T + (420.5 + 728. i)T^{2} \) | |
| 31 | \( 1 + (4.74 + 8.21i)T + (-480.5 + 832. i)T^{2} \) | |
| 37 | \( 1 + 52.9T + 1.36e3T^{2} \) | |
| 41 | \( 1 + (54.4 - 31.4i)T + (840.5 - 1.45e3i)T^{2} \) | |
| 43 | \( 1 + (-1.48 + 2.57i)T + (-924.5 - 1.60e3i)T^{2} \) | |
| 47 | \( 1 + (-65.7 - 37.9i)T + (1.10e3 + 1.91e3i)T^{2} \) | |
| 53 | \( 1 + 2.91iT - 2.80e3T^{2} \) | |
| 59 | \( 1 + (-52.9 + 30.5i)T + (1.74e3 - 3.01e3i)T^{2} \) | |
| 61 | \( 1 + (59.9 - 103. i)T + (-1.86e3 - 3.22e3i)T^{2} \) | |
| 67 | \( 1 + (20.5 + 35.5i)T + (-2.24e3 + 3.88e3i)T^{2} \) | |
| 71 | \( 1 + 10.2iT - 5.04e3T^{2} \) | |
| 73 | \( 1 + 20.0T + 5.32e3T^{2} \) | |
| 79 | \( 1 + (20.0 - 34.6i)T + (-3.12e3 - 5.40e3i)T^{2} \) | |
| 83 | \( 1 + (94.0 + 54.2i)T + (3.44e3 + 5.96e3i)T^{2} \) | |
| 89 | \( 1 - 106. iT - 7.92e3T^{2} \) | |
| 97 | \( 1 + (-28.4 + 49.2i)T + (-4.70e3 - 8.14e3i)T^{2} \) | |
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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
−9.673825516766000830209661668182, −8.936320287388875828241510982706, −8.501402526113183822523177305174, −7.09957381441789981244154342877, −6.72144921675559747456445890619, −5.87004424972439886724527500961, −4.60569857612560090724702434350, −3.69113727409235724577798575875, −2.89413816076713285371743147026, −1.57909122588984928344423488934, 0.44650130575570968005827899992, 1.05169544729304689585575791122, 3.34607388418964566647023002784, 3.63322920158371041256197663834, 4.61699245940519929248246468697, 5.76781095445020851997192161530, 6.79143511959540274422044157257, 7.38568066087644662855389441346, 8.350627854094044967711309154681, 8.894058826044726384405926454583