Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 105
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.36e15·2-s − 8.33e23·3-s + 1.37e31·4-s + 6.83e36·5-s + 6.14e39·6-s + 4.41e44·7-s + 1.97e47·8-s − 1.24e50·9-s − 5.03e52·10-s − 4.48e54·11-s − 1.14e55·12-s + 1.91e58·13-s − 3.25e60·14-s − 5.69e60·15-s − 2.01e63·16-s + 1.58e64·17-s + 9.17e65·18-s − 2.66e67·19-s + 9.38e67·20-s − 3.68e68·21-s + 3.30e70·22-s − 2.81e71·23-s − 1.64e71·24-s + 2.20e73·25-s − 1.41e74·26-s + 2.08e74·27-s + 6.06e75·28-s + ⋯
L(s)  = 1  − 1.15·2-s − 0.0745·3-s + 0.338·4-s + 1.37·5-s + 0.0861·6-s + 1.89·7-s + 0.765·8-s − 0.994·9-s − 1.59·10-s − 0.951·11-s − 0.0252·12-s + 0.632·13-s − 2.19·14-s − 0.102·15-s − 1.22·16-s + 0.398·17-s + 1.15·18-s − 1.95·19-s + 0.465·20-s − 0.141·21-s + 1.10·22-s − 0.910·23-s − 0.0570·24-s + 0.894·25-s − 0.731·26-s + 0.148·27-s + 0.641·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(106-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+52.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(105\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1,\ (\ :105/2),\ -1)\)
\(L(53)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{107}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where,\(F_p(T)\) is a polynomial of degree 2.
$p$$F_p(T)$
good2 \( 1 + 7.36e15T + 4.05e31T^{2} \)
3 \( 1 + 8.33e23T + 1.25e50T^{2} \)
5 \( 1 - 6.83e36T + 2.46e73T^{2} \)
7 \( 1 - 4.41e44T + 5.43e88T^{2} \)
11 \( 1 + 4.48e54T + 2.21e109T^{2} \)
13 \( 1 - 1.91e58T + 9.20e116T^{2} \)
17 \( 1 - 1.58e64T + 1.57e129T^{2} \)
19 \( 1 + 2.66e67T + 1.85e134T^{2} \)
23 \( 1 + 2.81e71T + 9.58e142T^{2} \)
29 \( 1 + 4.39e76T + 3.56e153T^{2} \)
31 \( 1 - 4.22e77T + 3.91e156T^{2} \)
37 \( 1 - 1.25e82T + 4.58e164T^{2} \)
41 \( 1 + 2.50e84T + 2.19e169T^{2} \)
43 \( 1 + 6.45e85T + 3.26e171T^{2} \)
47 \( 1 + 3.82e87T + 3.71e175T^{2} \)
53 \( 1 + 3.23e90T + 1.11e181T^{2} \)
59 \( 1 + 3.76e92T + 8.69e185T^{2} \)
61 \( 1 + 9.74e92T + 2.88e187T^{2} \)
67 \( 1 + 4.90e95T + 5.46e191T^{2} \)
71 \( 1 + 2.06e96T + 2.41e194T^{2} \)
73 \( 1 + 4.35e97T + 4.45e195T^{2} \)
79 \( 1 + 6.86e99T + 1.78e199T^{2} \)
83 \( 1 - 1.96e100T + 3.18e201T^{2} \)
89 \( 1 - 1.19e102T + 4.85e204T^{2} \)
97 \( 1 - 1.02e104T + 4.08e208T^{2} \)
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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

−13.47417813273510230722423712576, −11.16138844740525083036379977109, −10.26937862063163483379321044986, −8.690642052838314911773466288106, −8.001687833896636221822622490901, −5.91799329045029277410298420706, −4.75110070302069520763097418506, −2.17973936048721177847900977072, −1.53414086645799293493227973877, 0, 1.53414086645799293493227973877, 2.17973936048721177847900977072, 4.75110070302069520763097418506, 5.91799329045029277410298420706, 8.001687833896636221822622490901, 8.690642052838314911773466288106, 10.26937862063163483379321044986, 11.16138844740525083036379977109, 13.47417813273510230722423712576

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