Dirichlet series
L(s) = 1 | + 4.09e3·2-s + 1.62e5·3-s + 1.67e7·4-s + 2.44e8·5-s + 6.67e8·6-s − 1.76e10·7-s + 6.87e10·8-s − 8.20e11·9-s + 1.00e12·10-s − 1.12e13·11-s + 2.73e12·12-s + 4.24e12·13-s − 7.20e13·14-s + 3.97e13·15-s + 2.81e14·16-s − 1.13e15·17-s − 3.36e15·18-s + 2.98e15·19-s + 4.09e15·20-s − 2.86e15·21-s − 4.60e16·22-s − 1.22e17·23-s + 1.11e16·24-s + 5.96e16·25-s + 1.73e16·26-s − 2.71e17·27-s − 2.95e17·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.176·3-s + 1/2·4-s + 0.447·5-s + 0.125·6-s − 0.480·7-s + 0.353·8-s − 0.968·9-s + 0.316·10-s − 1.07·11-s + 0.0884·12-s + 0.0505·13-s − 0.339·14-s + 0.0791·15-s + 1/4·16-s − 0.473·17-s − 0.684·18-s + 0.309·19-s + 0.223·20-s − 0.0850·21-s − 0.763·22-s − 1.16·23-s + 0.0625·24-s + 1/5·25-s + 0.0357·26-s − 0.348·27-s − 0.240·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
Degree: | \(2\) |
Conductor: | \(10\) = \(2 \cdot 5\) |
Sign: | $-1$ |
Analytic conductor: | \(39.5996\) |
Root analytic conductor: | \(6.29282\) |
Motivic weight: | \(25\) |
Rational: | yes |
Arithmetic: | yes |
Character: | Trivial |
Primitive: | yes |
Self-dual: | yes |
Analytic rank: | \(1\) |
Selberg data: | \((2,\ 10,\ (\ :25/2),\ -1)\) |
Particular Values
\(L(13)\) | \(=\) | \(0\) |
\(L(\frac12)\) | \(=\) | \(0\) |
\(L(\frac{27}{2})\) | not available | |
\(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$ | $F_p(T)$ | |
---|---|---|
bad | 2 | \( 1 - p^{12} T \) |
5 | \( 1 - p^{12} T \) | |
good | 3 | \( 1 - 6032 p^{3} T + p^{25} T^{2} \) |
7 | \( 1 + 359201908 p^{2} T + p^{25} T^{2} \) | |
11 | \( 1 + 1021852166868 p T + p^{25} T^{2} \) | |
13 | \( 1 - 4242208935134 T + p^{25} T^{2} \) | |
17 | \( 1 + 1137855948691902 T + p^{25} T^{2} \) | |
19 | \( 1 - 157360283511980 p T + p^{25} T^{2} \) | |
23 | \( 1 + 5317628493231252 p T + p^{25} T^{2} \) | |
29 | \( 1 + 2120475579683207970 T + p^{25} T^{2} \) | |
31 | \( 1 - 4225863091688971352 T + p^{25} T^{2} \) | |
37 | \( 1 - 24167876952768085478 T + p^{25} T^{2} \) | |
41 | \( 1 + \)\(16\!\cdots\!98\)\( T + p^{25} T^{2} \) | |
43 | \( 1 + \)\(31\!\cdots\!56\)\( T + p^{25} T^{2} \) | |
47 | \( 1 + \)\(12\!\cdots\!32\)\( T + p^{25} T^{2} \) | |
53 | \( 1 + \)\(47\!\cdots\!86\)\( T + p^{25} T^{2} \) | |
59 | \( 1 - \)\(12\!\cdots\!60\)\( T + p^{25} T^{2} \) | |
61 | \( 1 + \)\(20\!\cdots\!98\)\( T + p^{25} T^{2} \) | |
67 | \( 1 - \)\(61\!\cdots\!48\)\( T + p^{25} T^{2} \) | |
71 | \( 1 - \)\(13\!\cdots\!52\)\( T + p^{25} T^{2} \) | |
73 | \( 1 - \)\(22\!\cdots\!54\)\( T + p^{25} T^{2} \) | |
79 | \( 1 - \)\(38\!\cdots\!80\)\( T + p^{25} T^{2} \) | |
83 | \( 1 + \)\(77\!\cdots\!76\)\( T + p^{25} T^{2} \) | |
89 | \( 1 + \)\(84\!\cdots\!10\)\( T + p^{25} T^{2} \) | |
97 | \( 1 + \)\(61\!\cdots\!82\)\( T + p^{25} T^{2} \) | |
show more | ||
show less |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.96338087411647758477151219056, −12.99026823822012865044309992434, −11.42808512915844142692547936198, −9.910323625347269051789617519540, −8.133918057647150443036575118108, −6.34378961112213217888323894926, −5.14461330758947329726578770829, −3.31296964318430204781834599520, −2.14553953613213796096173435862, 0, 2.14553953613213796096173435862, 3.31296964318430204781834599520, 5.14461330758947329726578770829, 6.34378961112213217888323894926, 8.133918057647150443036575118108, 9.910323625347269051789617519540, 11.42808512915844142692547936198, 12.99026823822012865044309992434, 13.96338087411647758477151219056