Properties

Label 2-1-1.1-c51-0-0
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  − 1.27e7·2-s − 5.15e11·3-s − 2.09e15·4-s − 3.83e17·5-s + 6.54e18·6-s − 2.02e21·7-s + 5.51e22·8-s − 1.88e24·9-s + 4.87e24·10-s − 2.47e25·11-s + 1.07e27·12-s − 1.34e27·13-s + 2.56e28·14-s + 1.97e29·15-s + 4.00e30·16-s + 3.03e31·17-s + 2.39e31·18-s + 7.39e32·19-s + 8.01e32·20-s + 1.04e33·21-s + 3.13e32·22-s − 7.84e34·23-s − 2.84e34·24-s − 2.96e35·25-s + 1.70e34·26-s + 2.08e36·27-s + 4.22e36·28-s + ⋯
L(s)  = 1  − 0.267·2-s − 0.351·3-s − 0.928·4-s − 0.575·5-s + 0.0939·6-s − 0.569·7-s + 0.516·8-s − 0.876·9-s + 0.154·10-s − 0.0687·11-s + 0.325·12-s − 0.0527·13-s + 0.152·14-s + 0.202·15-s + 0.790·16-s + 1.27·17-s + 0.234·18-s + 1.82·19-s + 0.534·20-s + 0.200·21-s + 0.0184·22-s − 1.48·23-s − 0.181·24-s − 0.668·25-s + 0.0141·26-s + 0.658·27-s + 0.529·28-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

\(L(26)\) \(\approx\) \(0.6668793110\)
\(L(\frac12)\) \(\approx\) \(0.6668793110\)
\(L(\frac{53}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 1.27e7T + 2.25e15T^{2} \)
3 \( 1 + 5.15e11T + 2.15e24T^{2} \)
5 \( 1 + 3.83e17T + 4.44e35T^{2} \)
7 \( 1 + 2.02e21T + 1.25e43T^{2} \)
11 \( 1 + 2.47e25T + 1.29e53T^{2} \)
13 \( 1 + 1.34e27T + 6.47e56T^{2} \)
17 \( 1 - 3.03e31T + 5.66e62T^{2} \)
19 \( 1 - 7.39e32T + 1.64e65T^{2} \)
23 \( 1 + 7.84e34T + 2.80e69T^{2} \)
29 \( 1 - 1.79e37T + 3.82e74T^{2} \)
31 \( 1 + 9.27e37T + 1.14e76T^{2} \)
37 \( 1 - 6.86e39T + 9.51e79T^{2} \)
41 \( 1 + 1.74e41T + 1.78e82T^{2} \)
43 \( 1 - 5.07e41T + 2.02e83T^{2} \)
47 \( 1 + 2.50e42T + 1.89e85T^{2} \)
53 \( 1 - 1.56e44T + 8.67e87T^{2} \)
59 \( 1 + 5.63e44T + 2.05e90T^{2} \)
61 \( 1 - 3.98e45T + 1.12e91T^{2} \)
67 \( 1 + 1.33e46T + 1.34e93T^{2} \)
71 \( 1 - 2.44e47T + 2.59e94T^{2} \)
73 \( 1 - 1.85e47T + 1.07e95T^{2} \)
79 \( 1 - 1.63e48T + 6.01e96T^{2} \)
83 \( 1 - 5.04e48T + 7.46e97T^{2} \)
89 \( 1 - 1.69e48T + 2.62e99T^{2} \)
97 \( 1 + 6.10e50T + 2.11e101T^{2} \)
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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

−19.74420216304625018629050449642, −18.09738673547305138168067273539, −16.37834763691635138467263785988, −14.03157738336776121271156052834, −11.99400602044885273180751543862, −9.792279032448167874248641477831, −7.961279974284214418241080204082, −5.48941031099175255744586727946, −3.51688233082781277606046266650, −0.62068469340329910125506647824, 0.62068469340329910125506647824, 3.51688233082781277606046266650, 5.48941031099175255744586727946, 7.961279974284214418241080204082, 9.792279032448167874248641477831, 11.99400602044885273180751543862, 14.03157738336776121271156052834, 16.37834763691635138467263785988, 18.09738673547305138168067273539, 19.74420216304625018629050449642

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