Properties

Degree $2$
Conductor $1$
Sign $-1$
Motivic weight $61$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.84e9·2-s − 3.79e14·3-s + 5.77e18·4-s − 6.89e20·5-s − 1.07e24·6-s − 9.80e25·7-s + 9.84e27·8-s + 1.68e28·9-s − 1.95e30·10-s − 2.74e31·11-s − 2.19e33·12-s + 2.74e32·13-s − 2.78e35·14-s + 2.61e35·15-s + 1.46e37·16-s − 2.27e37·17-s + 4.80e37·18-s − 1.42e39·19-s − 3.97e39·20-s + 3.72e40·21-s − 7.81e40·22-s + 1.13e41·23-s − 3.73e42·24-s − 3.86e42·25-s + 7.80e41·26-s + 4.18e43·27-s − 5.65e44·28-s + ⋯
L(s)  = 1  + 1.87·2-s − 1.06·3-s + 2.50·4-s − 0.331·5-s − 1.99·6-s − 1.64·7-s + 2.81·8-s + 0.132·9-s − 0.619·10-s − 0.474·11-s − 2.66·12-s + 0.0290·13-s − 3.07·14-s + 0.352·15-s + 2.75·16-s − 0.673·17-s + 0.248·18-s − 1.42·19-s − 0.828·20-s + 1.74·21-s − 0.888·22-s + 0.332·23-s − 2.99·24-s − 0.890·25-s + 0.0543·26-s + 0.922·27-s − 4.11·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(62-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+61/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Motivic weight: \(61\)
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :61/2),\ -1)\)

Particular Values

\(L(31)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{63}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 2.84e9T + 2.30e18T^{2} \)
3 \( 1 + 3.79e14T + 1.27e29T^{2} \)
5 \( 1 + 6.89e20T + 4.33e42T^{2} \)
7 \( 1 + 9.80e25T + 3.55e51T^{2} \)
11 \( 1 + 2.74e31T + 3.34e63T^{2} \)
13 \( 1 - 2.74e32T + 8.92e67T^{2} \)
17 \( 1 + 2.27e37T + 1.14e75T^{2} \)
19 \( 1 + 1.42e39T + 1.00e78T^{2} \)
23 \( 1 - 1.13e41T + 1.16e83T^{2} \)
29 \( 1 - 2.20e44T + 1.60e89T^{2} \)
31 \( 1 + 7.87e44T + 9.39e90T^{2} \)
37 \( 1 - 5.41e47T + 4.57e95T^{2} \)
41 \( 1 - 3.44e48T + 2.39e98T^{2} \)
43 \( 1 + 2.45e49T + 4.38e99T^{2} \)
47 \( 1 - 7.88e50T + 9.95e101T^{2} \)
53 \( 1 - 4.85e52T + 1.51e105T^{2} \)
59 \( 1 + 1.37e54T + 1.05e108T^{2} \)
61 \( 1 - 4.54e53T + 8.03e108T^{2} \)
67 \( 1 + 8.79e55T + 2.45e111T^{2} \)
71 \( 1 - 3.27e56T + 8.44e112T^{2} \)
73 \( 1 + 6.07e56T + 4.59e113T^{2} \)
79 \( 1 - 1.23e58T + 5.69e115T^{2} \)
83 \( 1 + 6.61e58T + 1.15e117T^{2} \)
89 \( 1 + 1.38e59T + 8.18e118T^{2} \)
97 \( 1 + 3.96e60T + 1.55e121T^{2} \)
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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

−16.57634577835638455759928324206, −15.42408694129414743263261836839, −13.26463982769579384643463080700, −12.21392220019893156749168549316, −10.75206702783945454322899220034, −6.70471629149157498504648306267, −5.80335275982875559328762633248, −4.21345880439988315458369248470, −2.72237056783497296800658086066, 0, 2.72237056783497296800658086066, 4.21345880439988315458369248470, 5.80335275982875559328762633248, 6.70471629149157498504648306267, 10.75206702783945454322899220034, 12.21392220019893156749168549316, 13.26463982769579384643463080700, 15.42408694129414743263261836839, 16.57634577835638455759928324206

Graph of the $Z$-function along the critical line