Dirichlet series
| L(s) = 1 | − 6.33e29·4-s − 3.68e40·7-s − 1.20e55·13-s + 4.01e59·16-s − 3.68e63·19-s − 1.57e69·25-s + 2.33e70·28-s − 1.28e74·31-s + 8.09e77·37-s − 1.22e81·43-s − 4.60e83·49-s + 7.64e84·52-s + 2.78e87·61-s − 2.54e89·64-s + 3.12e90·67-s − 2.77e92·73-s + 2.33e93·76-s − 7.87e93·79-s + 4.44e95·91-s − 1.20e98·97-s + 9.99e98·100-s − 1.48e100·112-s + ⋯ |
| L(s) = 1 | − 4-s − 0.0542·7-s − 0.873·13-s + 16-s − 1.85·19-s − 25-s + 0.0542·28-s − 1.93·31-s + 1.91·37-s − 1.70·43-s − 0.997·49-s + 0.873·52-s + 0.117·61-s − 64-s + 1.26·67-s − 1.61·73-s + 1.85·76-s − 0.920·79-s + 0.0473·91-s − 0.543·97-s + 100-s − 0.0542·112-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(100-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+99/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(9\) = \(3^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(558.609\) |
| Root analytic conductor: | \(23.6349\) |
| Motivic weight: | \(99\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((2,\ 9,\ (\ :99/2),\ 1)\) |
Particular Values
| \(L(50)\) | \(\approx\) | \(0.02240953788\) |
| \(L(\frac12)\) | \(\approx\) | \(0.02240953788\) |
| \(L(\frac{101}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + p^{99} T^{2} \) |
| 5 | \( 1 + p^{99} T^{2} \) | |
| 7 | \( 1 + \)\(36\!\cdots\!40\)\( T + p^{99} T^{2} \) | |
| 11 | \( 1 + p^{99} T^{2} \) | |
| 13 | \( 1 + \)\(12\!\cdots\!10\)\( T + p^{99} T^{2} \) | |
| 17 | \( 1 + p^{99} T^{2} \) | |
| 19 | \( 1 + \)\(36\!\cdots\!04\)\( T + p^{99} T^{2} \) | |
| 23 | \( 1 + p^{99} T^{2} \) | |
| 29 | \( 1 + p^{99} T^{2} \) | |
| 31 | \( 1 + \)\(12\!\cdots\!72\)\( T + p^{99} T^{2} \) | |
| 37 | \( 1 - \)\(80\!\cdots\!30\)\( T + p^{99} T^{2} \) | |
| 41 | \( 1 + p^{99} T^{2} \) | |
| 43 | \( 1 + \)\(12\!\cdots\!60\)\( T + p^{99} T^{2} \) | |
| 47 | \( 1 + p^{99} T^{2} \) | |
| 53 | \( 1 + p^{99} T^{2} \) | |
| 59 | \( 1 + p^{99} T^{2} \) | |
| 61 | \( 1 - \)\(27\!\cdots\!42\)\( T + p^{99} T^{2} \) | |
| 67 | \( 1 - \)\(31\!\cdots\!60\)\( T + p^{99} T^{2} \) | |
| 71 | \( 1 + p^{99} T^{2} \) | |
| 73 | \( 1 + \)\(27\!\cdots\!30\)\( T + p^{99} T^{2} \) | |
| 79 | \( 1 + \)\(78\!\cdots\!76\)\( T + p^{99} T^{2} \) | |
| 83 | \( 1 + p^{99} T^{2} \) | |
| 89 | \( 1 + p^{99} T^{2} \) | |
| 97 | \( 1 + \)\(12\!\cdots\!90\)\( T + p^{99} T^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.430137989404172297282633781320, −8.480663346729351050966295307705, −7.60123708549594464943599756800, −6.35665994266684769038950329418, −5.33763996812690099983080450304, −4.42922126326569788162779684247, −3.69703250675794071184371433978, −2.45034939179629211578981378167, −1.47707628857318269078935873942, −0.05194598604784005219152009507, 0.05194598604784005219152009507, 1.47707628857318269078935873942, 2.45034939179629211578981378167, 3.69703250675794071184371433978, 4.42922126326569788162779684247, 5.33763996812690099983080450304, 6.35665994266684769038950329418, 7.60123708549594464943599756800, 8.480663346729351050966295307705, 9.430137989404172297282633781320