Dirichlet series
| L(s) = 1 | − 2-s − 3·3-s − 5-s + 3·6-s − 6·7-s + 8-s + 4·9-s + 10-s − 13-s + 6·14-s + 3·15-s − 16-s + 2·17-s − 4·18-s + 18·21-s − 2·23-s − 3·24-s − 6·25-s + 26-s + 29-s − 3·30-s − 31-s − 2·34-s + 6·35-s − 4·37-s + 3·39-s − 40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 0.447·5-s + 1.22·6-s − 2.26·7-s + 0.353·8-s + 4/3·9-s + 0.316·10-s − 0.277·13-s + 1.60·14-s + 0.774·15-s − 1/4·16-s + 0.485·17-s − 0.942·18-s + 3.92·21-s − 0.417·23-s − 0.612·24-s − 6/5·25-s + 0.196·26-s + 0.185·29-s − 0.547·30-s − 0.179·31-s − 0.342·34-s + 1.01·35-s − 0.657·37-s + 0.480·39-s − 0.158·40-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 2172 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2172 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(2172\) = \(2^{2} \cdot 3 \cdot 181\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.138488\) |
| Root analytic conductor: | \(0.610033\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 2172,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_2$ | \( 1 + T + T^{2} \) | |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) | ||
| 181 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 7 T + p T^{2} ) \) | ||
| good | 5 | $D_{4}$ | \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.5.b_h |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.7.g_t | |
| 11 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) | 2.11.a_ar | |
| 13 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.13.b_c | |
| 17 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.17.ac_e | |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.19.a_k | |
| 23 | $D_{4}$ | \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.23.c_t | |
| 29 | $D_{4}$ | \( 1 - T - 5 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.29.ab_af | |
| 31 | $D_{4}$ | \( 1 + T + 23 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.31.b_x | |
| 37 | $D_{4}$ | \( 1 + 4 T + 17 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.37.e_r | |
| 41 | $D_{4}$ | \( 1 - 5 T + 7 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.41.af_h | |
| 43 | $D_{4}$ | \( 1 + 4 T + 62 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.43.e_ck | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.47.ad_bo | |
| 53 | $D_{4}$ | \( 1 + T + 40 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.53.b_bo | |
| 59 | $D_{4}$ | \( 1 + 4 T + 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.59.e_br | |
| 61 | $D_{4}$ | \( 1 + 10 T + 134 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.61.k_fe | |
| 67 | $D_{4}$ | \( 1 - 3 T + 85 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.67.ad_dh | |
| 71 | $D_{4}$ | \( 1 - 3 T - 56 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.71.ad_ace | |
| 73 | $D_{4}$ | \( 1 + 12 T + 148 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.73.m_fs | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.79.g_eo | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) | 2.83.k_cs | |
| 89 | $D_{4}$ | \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.89.e_fm | |
| 97 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) | 2.97.a_t | |
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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
−18.8824552440, −18.4861427438, −17.8583197664, −17.2201716610, −17.0059817625, −16.3658681163, −15.9067260155, −15.8320060284, −14.8766984516, −13.8825183266, −13.3452552832, −12.5011817047, −12.3686722539, −11.7005874097, −11.0221712450, −10.3373513707, −9.86191513787, −9.43418835851, −8.49191017900, −7.46130316461, −6.85502420859, −6.07691672544, −5.67563813407, −4.41195112992, −3.27425598324, 0, 3.27425598324, 4.41195112992, 5.67563813407, 6.07691672544, 6.85502420859, 7.46130316461, 8.49191017900, 9.43418835851, 9.86191513787, 10.3373513707, 11.0221712450, 11.7005874097, 12.3686722539, 12.5011817047, 13.3452552832, 13.8825183266, 14.8766984516, 15.8320060284, 15.9067260155, 16.3658681163, 17.0059817625, 17.2201716610, 17.8583197664, 18.4861427438, 18.8824552440