Error: table mf_hecke_newspace_traces does not exist
Dirichlet series
| L(s) = 1 | + (−0.309 + 0.951i)2-s + (−2.33 + 1.69i)3-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + (−0.890 − 2.74i)6-s + (−3.17 − 2.30i)7-s + (0.809 − 0.587i)8-s + (1.64 − 5.05i)9-s − 0.999·10-s + (−1.42 + 2.99i)11-s + 2.88·12-s + (−1.80 + 5.54i)13-s + (3.17 − 2.30i)14-s + (−2.33 − 1.69i)15-s + (0.309 + 0.951i)16-s + (0.332 + 1.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (−1.34 + 0.978i)3-s + (−0.404 − 0.293i)4-s + (0.138 + 0.425i)5-s + (−0.363 − 1.11i)6-s + (−1.20 − 0.873i)7-s + (0.286 − 0.207i)8-s + (0.547 − 1.68i)9-s − 0.316·10-s + (−0.430 + 0.902i)11-s + 0.832·12-s + (−0.499 + 1.53i)13-s + (0.849 − 0.617i)14-s + (−0.602 − 0.437i)15-s + (0.0772 + 0.237i)16-s + (0.0806 + 0.248i)17-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.227i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(110\) = \(2 \cdot 5 \cdot 11\) |
| Sign: | $-0.973 + 0.227i$ |
| Analytic conductor: | \(0.878354\) |
| Root analytic conductor: | \(0.937205\) |
| Motivic weight: | \(1\) |
| Rational: | no |
| Arithmetic: | yes |
| Character: | $\chi_{110} (31, \cdot )$ |
| Primitive: | yes |
| Self-dual: | no |
| Analytic rank: | \(0\) |
| Selberg data: | \((2,\ 110,\ (\ :1/2),\ -0.973 + 0.227i)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.0392920 - 0.341234i\) |
| \(L(\frac12)\) | \(\approx\) | \(0.0392920 - 0.341234i\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (-0.309 - 0.951i)T \) | |
| 11 | \( 1 + (1.42 - 2.99i)T \) | |
| good | 3 | \( 1 + (2.33 - 1.69i)T + (0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + (3.17 + 2.30i)T + (2.16 + 6.65i)T^{2} \) | |
| 13 | \( 1 + (1.80 - 5.54i)T + (-10.5 - 7.64i)T^{2} \) | |
| 17 | \( 1 + (-0.332 - 1.02i)T + (-13.7 + 9.99i)T^{2} \) | |
| 19 | \( 1 + (1.22 - 0.891i)T + (5.87 - 18.0i)T^{2} \) | |
| 23 | \( 1 - 5.07T + 23T^{2} \) | |
| 29 | \( 1 + (0.306 + 0.222i)T + (8.96 + 27.5i)T^{2} \) | |
| 31 | \( 1 + (2.02 - 6.24i)T + (-25.0 - 18.2i)T^{2} \) | |
| 37 | \( 1 + (-0.832 - 0.604i)T + (11.4 + 35.1i)T^{2} \) | |
| 41 | \( 1 + (3.19 - 2.32i)T + (12.6 - 38.9i)T^{2} \) | |
| 43 | \( 1 + 0.446T + 43T^{2} \) | |
| 47 | \( 1 + (-6.14 + 4.46i)T + (14.5 - 44.6i)T^{2} \) | |
| 53 | \( 1 + (-1.93 + 5.95i)T + (-42.8 - 31.1i)T^{2} \) | |
| 59 | \( 1 + (-8.89 - 6.46i)T + (18.2 + 56.1i)T^{2} \) | |
| 61 | \( 1 + (3.10 + 9.54i)T + (-49.3 + 35.8i)T^{2} \) | |
| 67 | \( 1 + 5.98T + 67T^{2} \) | |
| 71 | \( 1 + (-1.55 - 4.78i)T + (-57.4 + 41.7i)T^{2} \) | |
| 73 | \( 1 + (2.86 + 2.07i)T + (22.5 + 69.4i)T^{2} \) | |
| 79 | \( 1 + (-0.218 + 0.671i)T + (-63.9 - 46.4i)T^{2} \) | |
| 83 | \( 1 + (-3.62 - 11.1i)T + (-67.1 + 48.7i)T^{2} \) | |
| 89 | \( 1 - 0.0446T + 89T^{2} \) | |
| 97 | \( 1 + (5.05 - 15.5i)T + (-78.4 - 57.0i)T^{2} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.52120799033476397019713266463, −13.19598093379782249647790055293, −12.03207215019985204174228450250, −10.73099866319874705828297254719, −10.05806322701886929500251765093, −9.296122734987927036152281778673, −7.04811719089829024307130350096, −6.55123500778011176458540553955, −5.10471948390995157179385128087, −3.98131011991580302443621802076, 0.47196513080877978362943346881, 2.80903151152349718685901009502, 5.32400956280722063658849616265, 6.02858581003208838941006143607, 7.50742988166640457580016879153, 8.891524582071811297802986665830, 10.19607486199793323714582944479, 11.18555570440107002720692737151, 12.21315723316123518466874980947, 12.87237821734910835654232683044