Dirichlet series
| L(s) = 1 | − 2-s + 4-s + 4·5-s + 2·7-s − 8-s − 4·10-s − 8·13-s − 2·14-s + 16-s − 4·17-s + 2·19-s + 4·20-s − 6·23-s + 3·25-s + 8·26-s + 2·28-s + 4·29-s + 2·31-s − 32-s + 4·34-s + 8·35-s + 2·37-s − 2·38-s − 4·40-s − 14·41-s − 22·43-s + 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.755·7-s − 0.353·8-s − 1.26·10-s − 2.21·13-s − 0.534·14-s + 1/4·16-s − 0.970·17-s + 0.458·19-s + 0.894·20-s − 1.25·23-s + 3/5·25-s + 1.56·26-s + 0.377·28-s + 0.742·29-s + 0.359·31-s − 0.176·32-s + 0.685·34-s + 1.35·35-s + 0.328·37-s − 0.324·38-s − 0.632·40-s − 2.18·41-s − 3.35·43-s + 0.884·46-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(373248\) = \(2^{9} \cdot 3^{6}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(23.7986\) |
| Root analytic conductor: | \(2.20870\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 373248,\ (\ :1/2, 1/2),\ -1)\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ | |
|---|---|---|---|---|
| bad | 2 | $C_1$ | \( 1 + T \) | |
| 3 | \( 1 \) | |||
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \) | 2.5.ae_n |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.7.ac_l | |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.11.a_n | |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | 2.13.i_bq | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.17.e_bi | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) | 2.19.ac_bm | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.23.g_bu | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.29.ae_bu | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.31.ac_bv | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) | 2.37.ac_cw | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.41.o_fa | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.43.w_hy | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.47.g_w | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.53.ae_cj | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) | 2.59.am_eo | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.61.ai_es | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) | 2.67.aba_lq | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.71.m_fm | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.73.ac_df | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.79.e_ck | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) | 2.83.m_hl | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) | 2.89.w_jq | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) | 2.97.k_hv | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.1830201225, −12.8214281252, −11.9598790235, −11.9118486001, −11.4654293297, −11.1128063228, −10.1152647782, −9.98703221781, −9.87204525884, −9.79241457837, −8.86377903873, −8.35941284278, −8.26804890939, −7.61059019835, −6.87645619133, −6.73860900675, −6.31538067807, −5.37363015538, −5.30469420615, −4.84789482589, −4.10619268148, −3.14627859790, −2.44850978333, −1.95251803353, −1.64550789318, 0, 1.64550789318, 1.95251803353, 2.44850978333, 3.14627859790, 4.10619268148, 4.84789482589, 5.30469420615, 5.37363015538, 6.31538067807, 6.73860900675, 6.87645619133, 7.61059019835, 8.26804890939, 8.35941284278, 8.86377903873, 9.79241457837, 9.87204525884, 9.98703221781, 10.1152647782, 11.1128063228, 11.4654293297, 11.9118486001, 11.9598790235, 12.8214281252, 13.1830201225