Properties

Degree 2
Conductor $ 1 $
Sign $1$
Motivic weight 103
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.46e15·2-s − 5.92e24·3-s + 1.97e31·4-s + 6.19e35·5-s − 3.23e40·6-s + 6.38e43·7-s + 5.24e46·8-s + 2.11e49·9-s + 3.38e51·10-s − 3.56e53·11-s − 1.16e56·12-s − 1.22e57·13-s + 3.48e59·14-s − 3.66e60·15-s + 8.64e61·16-s + 1.95e63·17-s + 1.15e65·18-s − 1.27e65·19-s + 1.22e67·20-s − 3.78e68·21-s − 1.94e69·22-s + 1.72e70·23-s − 3.10e71·24-s − 6.02e71·25-s − 6.71e72·26-s − 4.28e73·27-s + 1.25e75·28-s + ⋯
L(s)  = 1  + 1.71·2-s − 1.58·3-s + 1.94·4-s + 0.624·5-s − 2.72·6-s + 1.91·7-s + 1.62·8-s + 1.52·9-s + 1.07·10-s − 0.832·11-s − 3.08·12-s − 0.526·13-s + 3.29·14-s − 0.990·15-s + 0.840·16-s + 0.838·17-s + 2.60·18-s − 0.177·19-s + 1.21·20-s − 3.04·21-s − 1.42·22-s + 1.28·23-s − 2.57·24-s − 0.610·25-s − 0.903·26-s − 0.825·27-s + 3.73·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \,\Lambda(104-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+51.5) \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(103\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1,\ (\ :103/2),\ 1)$
$L(52)$  $\approx$  $5.130024776$
$L(\frac12)$  $\approx$  $5.130024776$
$L(\frac{105}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, \(F_p\) is a polynomial of degree 2.
$p$$F_p$
good2 \( 1 - 5.46e15T + 1.01e31T^{2} \)
3 \( 1 + 5.92e24T + 1.39e49T^{2} \)
5 \( 1 - 6.19e35T + 9.86e71T^{2} \)
7 \( 1 - 6.38e43T + 1.10e87T^{2} \)
11 \( 1 + 3.56e53T + 1.83e107T^{2} \)
13 \( 1 + 1.22e57T + 5.44e114T^{2} \)
17 \( 1 - 1.95e63T + 5.44e126T^{2} \)
19 \( 1 + 1.27e65T + 5.14e131T^{2} \)
23 \( 1 - 1.72e70T + 1.81e140T^{2} \)
29 \( 1 - 5.38e74T + 4.23e150T^{2} \)
31 \( 1 - 4.50e76T + 4.07e153T^{2} \)
37 \( 1 - 2.11e80T + 3.34e161T^{2} \)
41 \( 1 - 1.19e83T + 1.30e166T^{2} \)
43 \( 1 - 1.22e84T + 1.76e168T^{2} \)
47 \( 1 - 1.73e85T + 1.68e172T^{2} \)
53 \( 1 - 8.89e88T + 3.98e177T^{2} \)
59 \( 1 + 2.03e91T + 2.49e182T^{2} \)
61 \( 1 + 2.35e91T + 7.74e183T^{2} \)
67 \( 1 + 5.83e93T + 1.21e188T^{2} \)
71 \( 1 + 1.13e94T + 4.78e190T^{2} \)
73 \( 1 - 1.69e96T + 8.36e191T^{2} \)
79 \( 1 - 4.56e96T + 2.85e195T^{2} \)
83 \( 1 + 4.40e98T + 4.62e197T^{2} \)
89 \( 1 + 4.93e99T + 6.12e200T^{2} \)
97 \( 1 + 2.79e101T + 4.33e204T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.99273043217341093958106051686, −12.52151692603870485405232149475, −11.50049739033189966257727898466, −10.62617081756562645730815663806, −7.46213121808066968779398199157, −5.89026299293215642689776450808, −5.17957704985091042196262653789, −4.51768705190897098073789439257, −2.37590588962699693311472937317, −1.07600817673398778255563222649, 1.07600817673398778255563222649, 2.37590588962699693311472937317, 4.51768705190897098073789439257, 5.17957704985091042196262653789, 5.89026299293215642689776450808, 7.46213121808066968779398199157, 10.62617081756562645730815663806, 11.50049739033189966257727898466, 12.52151692603870485405232149475, 13.99273043217341093958106051686

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