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  − 3.84e15·2-s + 3.76e24·3-s + 4.63e30·4-s + 1.31e36·5-s − 1.44e40·6-s − 5.54e43·7-s + 2.11e46·8-s + 2.50e47·9-s − 5.03e51·10-s − 6.51e53·11-s + 1.74e55·12-s − 1.37e57·13-s + 2.13e59·14-s + 4.93e60·15-s − 1.28e62·16-s + 1.56e63·17-s − 9.61e62·18-s + 5.72e65·19-s + 6.07e66·20-s − 2.08e68·21-s + 2.50e69·22-s + 8.74e69·23-s + 7.96e70·24-s + 7.32e71·25-s + 5.27e72·26-s − 5.14e73·27-s − 2.56e74·28-s + ⋯
L(s)  = 1  − 1.20·2-s + 1.00·3-s + 0.457·4-s + 1.32·5-s − 1.21·6-s − 1.66·7-s + 0.655·8-s + 0.0179·9-s − 1.59·10-s − 1.52·11-s + 0.461·12-s − 0.588·13-s + 2.00·14-s + 1.33·15-s − 1.24·16-s + 0.672·17-s − 0.0217·18-s + 0.798·19-s + 0.603·20-s − 1.67·21-s + 1.83·22-s + 0.649·23-s + 0.661·24-s + 0.742·25-s + 0.709·26-s − 0.990·27-s − 0.760·28-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  =  $(2,\ 1,\ (\ :103/2),\ 1)$
$L(52)$  $\approx$  $1.206413790$
$L(\frac12)$  $\approx$  $1.206413790$
$L(\frac{105}{2})$   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 \( 1 + 3.84e15T + 1.01e31T^{2} \)
3 \( 1 - 3.76e24T + 1.39e49T^{2} \)
5 \( 1 - 1.31e36T + 9.86e71T^{2} \)
7 \( 1 + 5.54e43T + 1.10e87T^{2} \)
11 \( 1 + 6.51e53T + 1.83e107T^{2} \)
13 \( 1 + 1.37e57T + 5.44e114T^{2} \)
17 \( 1 - 1.56e63T + 5.44e126T^{2} \)
19 \( 1 - 5.72e65T + 5.14e131T^{2} \)
23 \( 1 - 8.74e69T + 1.81e140T^{2} \)
29 \( 1 - 1.45e75T + 4.23e150T^{2} \)
31 \( 1 - 4.89e76T + 4.07e153T^{2} \)
37 \( 1 + 3.88e80T + 3.34e161T^{2} \)
41 \( 1 + 1.41e83T + 1.30e166T^{2} \)
43 \( 1 - 2.24e84T + 1.76e168T^{2} \)
47 \( 1 + 7.36e85T + 1.68e172T^{2} \)
53 \( 1 - 7.09e88T + 3.98e177T^{2} \)
59 \( 1 - 2.12e91T + 2.49e182T^{2} \)
61 \( 1 - 9.84e91T + 7.74e183T^{2} \)
67 \( 1 - 7.24e93T + 1.21e188T^{2} \)
71 \( 1 + 2.66e93T + 4.78e190T^{2} \)
73 \( 1 - 5.38e95T + 8.36e191T^{2} \)
79 \( 1 + 5.00e97T + 2.85e195T^{2} \)
83 \( 1 + 9.52e98T + 4.62e197T^{2} \)
89 \( 1 + 2.31e100T + 6.12e200T^{2} \)
97 \( 1 - 9.94e101T + 4.33e204T^{2} \)
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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

−13.88272786057470900805287288305, −13.03923958001777027045492292685, −10.08770903883767053060467802962, −9.755068790422270160518222006983, −8.553041147218387855080960758225, −7.15367582618336961218050753907, −5.48141670592833357458724129920, −3.04358452573765442530295664594, −2.27547987401573463435651504036, −0.63635881128021820230484115981, 0.63635881128021820230484115981, 2.27547987401573463435651504036, 3.04358452573765442530295664594, 5.48141670592833357458724129920, 7.15367582618336961218050753907, 8.553041147218387855080960758225, 9.755068790422270160518222006983, 10.08770903883767053060467802962, 13.03923958001777027045492292685, 13.88272786057470900805287288305

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