Properties

Label 2-1-1.1-c51-0-2
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $16.4731$
Root an. cond. $4.05871$
Motivic weight $51$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.12e7·2-s − 2.41e12·3-s + 4.34e15·4-s + 2.42e17·5-s − 1.95e20·6-s + 2.85e21·7-s + 1.70e23·8-s + 3.66e24·9-s + 1.97e25·10-s + 3.52e26·11-s − 1.04e28·12-s + 3.67e28·13-s + 2.32e29·14-s − 5.85e29·15-s + 4.04e30·16-s − 6.54e30·17-s + 2.97e32·18-s + 2.55e32·19-s + 1.05e33·20-s − 6.88e33·21-s + 2.86e34·22-s − 1.69e34·23-s − 4.10e35·24-s − 3.85e35·25-s + 2.98e36·26-s − 3.63e36·27-s + 1.24e37·28-s + ⋯
L(s)  = 1  + 1.71·2-s − 1.64·3-s + 1.93·4-s + 0.364·5-s − 2.81·6-s + 0.804·7-s + 1.59·8-s + 1.69·9-s + 0.623·10-s + 0.979·11-s − 3.17·12-s + 1.44·13-s + 1.37·14-s − 0.598·15-s + 0.797·16-s − 0.274·17-s + 2.91·18-s + 0.630·19-s + 0.703·20-s − 1.32·21-s + 1.67·22-s − 0.320·23-s − 2.61·24-s − 0.867·25-s + 2.47·26-s − 1.15·27-s + 1.55·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(52-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+51/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(16.4731\)
Root analytic conductor: \(4.05871\)
Motivic weight: \(51\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :51/2),\ 1)\)

Particular Values

\(L(26)\) \(\approx\) \(3.703922825\)
\(L(\frac12)\) \(\approx\) \(3.703922825\)
\(L(\frac{53}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 8.12e7T + 2.25e15T^{2} \)
3 \( 1 + 2.41e12T + 2.15e24T^{2} \)
5 \( 1 - 2.42e17T + 4.44e35T^{2} \)
7 \( 1 - 2.85e21T + 1.25e43T^{2} \)
11 \( 1 - 3.52e26T + 1.29e53T^{2} \)
13 \( 1 - 3.67e28T + 6.47e56T^{2} \)
17 \( 1 + 6.54e30T + 5.66e62T^{2} \)
19 \( 1 - 2.55e32T + 1.64e65T^{2} \)
23 \( 1 + 1.69e34T + 2.80e69T^{2} \)
29 \( 1 - 3.94e36T + 3.82e74T^{2} \)
31 \( 1 - 9.38e37T + 1.14e76T^{2} \)
37 \( 1 + 1.74e40T + 9.51e79T^{2} \)
41 \( 1 - 2.14e41T + 1.78e82T^{2} \)
43 \( 1 + 2.13e41T + 2.02e83T^{2} \)
47 \( 1 - 4.04e42T + 1.89e85T^{2} \)
53 \( 1 + 4.94e42T + 8.67e87T^{2} \)
59 \( 1 + 1.20e45T + 2.05e90T^{2} \)
61 \( 1 - 2.20e45T + 1.12e91T^{2} \)
67 \( 1 - 4.56e46T + 1.34e93T^{2} \)
71 \( 1 + 1.91e47T + 2.59e94T^{2} \)
73 \( 1 + 8.49e46T + 1.07e95T^{2} \)
79 \( 1 - 2.35e48T + 6.01e96T^{2} \)
83 \( 1 - 5.38e48T + 7.46e97T^{2} \)
89 \( 1 + 7.58e49T + 2.62e99T^{2} \)
97 \( 1 + 8.33e50T + 2.11e101T^{2} \)
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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

−21.03901928242761506484149724951, −17.64239894597622728661864503175, −15.87708376043521940739686729965, −13.83094411907150913716270458144, −12.03795238952520272059363439468, −11.06266090259761044962942321159, −6.48736573311957801296620234338, −5.47892337175584191873993891140, −4.10763182976580858574694507598, −1.40440026423039880609375628767, 1.40440026423039880609375628767, 4.10763182976580858574694507598, 5.47892337175584191873993891140, 6.48736573311957801296620234338, 11.06266090259761044962942321159, 12.03795238952520272059363439468, 13.83094411907150913716270458144, 15.87708376043521940739686729965, 17.64239894597622728661864503175, 21.03901928242761506484149724951

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