Dirichlet series
| L(s) = 1 | − 2·2-s − 4·3-s + 2·4-s − 2·5-s + 8·6-s − 4·7-s + 7·9-s + 4·10-s − 8·12-s + 4·13-s + 8·14-s + 8·15-s − 4·16-s − 8·17-s − 14·18-s + 4·19-s − 4·20-s + 16·21-s − 3·25-s − 8·26-s − 4·27-s − 8·28-s − 8·29-s − 16·30-s + 8·32-s + 16·34-s + 8·35-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 4-s − 0.894·5-s + 3.26·6-s − 1.51·7-s + 7/3·9-s + 1.26·10-s − 2.30·12-s + 1.10·13-s + 2.13·14-s + 2.06·15-s − 16-s − 1.94·17-s − 3.29·18-s + 0.917·19-s − 0.894·20-s + 3.49·21-s − 3/5·25-s − 1.56·26-s − 0.769·27-s − 1.51·28-s − 1.48·29-s − 2.92·30-s + 1.41·32-s + 2.74·34-s + 1.35·35-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(968\) = \(2^{3} \cdot 11^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.0617205\) |
| Root analytic conductor: | \(0.498433\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 968,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_2$ | \( 1 + p T + p T^{2} \) | |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | ||
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) | 2.3.e_j |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.5.c_h | |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | 2.7.e_s | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.13.ae_ba | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.17.i_bu | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.19.ae_bm | |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.23.a_bt | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.29.i_cg | |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.31.a_n | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.37.ac_ct | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.41.e_by | |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.43.a_by | |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.47.a_be | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.53.e_dq | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.59.ae_ej | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) | 2.61.aq_go | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.67.m_gn | |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.71.a_fd | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) | 2.73.au_ic | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.79.i_fi | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.83.i_gw | |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) | 2.89.abe_pn | |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) | 2.97.o_jj | |
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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
−19.3981440797, −19.1857249719, −18.4484037676, −17.9414335735, −17.6304228292, −17.0336103204, −16.4150110915, −16.0208509362, −15.9140726033, −15.1389643255, −13.5686390571, −13.1644289392, −12.3870421342, −11.4512586103, −11.3646694668, −10.8920468072, −10.0355090972, −9.40167023369, −8.60353961929, −7.57356519541, −6.61017530637, −6.36261389471, −5.26718062024, −3.92131200413, 0, 3.92131200413, 5.26718062024, 6.36261389471, 6.61017530637, 7.57356519541, 8.60353961929, 9.40167023369, 10.0355090972, 10.8920468072, 11.3646694668, 11.4512586103, 12.3870421342, 13.1644289392, 13.5686390571, 15.1389643255, 15.9140726033, 16.0208509362, 16.4150110915, 17.0336103204, 17.6304228292, 17.9414335735, 18.4484037676, 19.1857249719, 19.3981440797