Dirichlet series
| L(s) = 1 | + 2·5-s + 4·13-s + 8·17-s − 25-s − 10·29-s + 12·37-s − 8·41-s − 7·49-s + 14·53-s + 12·61-s + 8·65-s + 6·73-s + 16·85-s − 16·89-s + 18·97-s − 2·101-s + 20·109-s − 16·113-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.10·13-s + 1.94·17-s − 1/5·25-s − 1.85·29-s + 1.97·37-s − 1.24·41-s − 49-s + 1.92·53-s + 1.53·61-s + 0.992·65-s + 0.702·73-s + 1.73·85-s − 1.69·89-s + 1.82·97-s − 0.199·101-s + 1.91·109-s − 1.50·113-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(2304\) = \(2^{8} \cdot 3^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(18.3975\) |
| Root analytic conductor: | \(4.28923\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((2,\ 2304,\ (\ :1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.369296459\) |
| \(L(\frac12)\) | \(\approx\) | \(2.369296459\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a | |
| 11 | \( 1 + p T^{2} \) | 1.11.a | |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae | |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai | |
| 19 | \( 1 + p T^{2} \) | 1.19.a | |
| 23 | \( 1 + p T^{2} \) | 1.23.a | |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k | |
| 31 | \( 1 + p T^{2} \) | 1.31.a | |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am | |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i | |
| 43 | \( 1 + p T^{2} \) | 1.43.a | |
| 47 | \( 1 + p T^{2} \) | 1.47.a | |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao | |
| 59 | \( 1 + p T^{2} \) | 1.59.a | |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am | |
| 67 | \( 1 + p T^{2} \) | 1.67.a | |
| 71 | \( 1 + p T^{2} \) | 1.71.a | |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag | |
| 79 | \( 1 + p T^{2} \) | 1.79.a | |
| 83 | \( 1 + p T^{2} \) | 1.83.a | |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q | |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as | |
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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
−9.112200135449134934371783246207, −8.202694220619429288756420575978, −7.55985426853550376471644781916, −6.54863571929054381318876237423, −5.71994566936884971110657383524, −5.39262958273976329743454417116, −4.01100966495944443006006263063, −3.27092596459393225052086143179, −2.05899636987229362790422468605, −1.07574057656802494464390490749, 1.07574057656802494464390490749, 2.05899636987229362790422468605, 3.27092596459393225052086143179, 4.01100966495944443006006263063, 5.39262958273976329743454417116, 5.71994566936884971110657383524, 6.54863571929054381318876237423, 7.55985426853550376471644781916, 8.202694220619429288756420575978, 9.112200135449134934371783246207