Dirichlet series
| L(s) = 1 | + 3·4-s − 2·5-s + 6·9-s − 8·11-s + 5·16-s − 2·19-s − 6·20-s − 25-s + 12·29-s − 8·31-s + 18·36-s − 20·41-s − 24·44-s − 12·45-s + 10·49-s + 16·55-s + 4·61-s + 3·64-s + 8·71-s − 6·76-s − 8·79-s − 10·80-s + 27·81-s + 4·89-s + 4·95-s − 48·99-s − 3·100-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 0.894·5-s + 2·9-s − 2.41·11-s + 5/4·16-s − 0.458·19-s − 1.34·20-s − 1/5·25-s + 2.22·29-s − 1.43·31-s + 3·36-s − 3.12·41-s − 3.61·44-s − 1.78·45-s + 10/7·49-s + 2.15·55-s + 0.512·61-s + 3/8·64-s + 0.949·71-s − 0.688·76-s − 0.900·79-s − 1.11·80-s + 3·81-s + 0.423·89-s + 0.410·95-s − 4.82·99-s − 0.299·100-s + ⋯ |
Functional equation
Invariants
| Degree: | \(4\) |
| Conductor: | \(9025\) = \(5^{2} \cdot 19^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.575441\) |
| Root analytic conductor: | \(0.870964\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 9025,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.150638088\) |
| \(L(\frac12)\) | \(\approx\) | \(1.150638088\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| bad | 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) | |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) | |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) | |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) | |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) | |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) | |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) | |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) | |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) | |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) | |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) | |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) | |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) | |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) | |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) | |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) | |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) | |
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Imaginary part of the first few zeros on the critical line
−14.39378607396707998210468115281, −13.33833719146884021116825649847, −13.28263545317051257194728846540, −12.59797454518394382825034810842, −12.02418282732934495801628845568, −11.84461485558858694293352140635, −10.70752799236204843250226126333, −10.68427162877309853596214878850, −10.25869934003058642049905339703, −9.644089965471797413239084841091, −8.336229919754813584904025088006, −8.105993347947847675102235226930, −7.32669114933864019521989753244, −7.08905699037559252250676843716, −6.50918098564082978345606367110, −5.41075217509758322553840415941, −4.78034076207371224866306495445, −3.84233590946015501174280589102, −2.85187722726909546411289531010, −1.91034861424110577446402056257, 1.91034861424110577446402056257, 2.85187722726909546411289531010, 3.84233590946015501174280589102, 4.78034076207371224866306495445, 5.41075217509758322553840415941, 6.50918098564082978345606367110, 7.08905699037559252250676843716, 7.32669114933864019521989753244, 8.105993347947847675102235226930, 8.336229919754813584904025088006, 9.644089965471797413239084841091, 10.25869934003058642049905339703, 10.68427162877309853596214878850, 10.70752799236204843250226126333, 11.84461485558858694293352140635, 12.02418282732934495801628845568, 12.59797454518394382825034810842, 13.28263545317051257194728846540, 13.33833719146884021116825649847, 14.39378607396707998210468115281