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  + 3.85e15·2-s − 1.61e25·3-s − 2.56e31·4-s + 9.75e36·5-s − 6.22e40·6-s − 2.29e44·7-s − 2.55e47·8-s + 1.34e50·9-s + 3.76e52·10-s + 1.09e54·11-s + 4.14e56·12-s − 1.87e58·13-s − 8.84e59·14-s − 1.57e62·15-s + 5.58e61·16-s + 5.30e64·17-s + 5.20e65·18-s + 1.09e67·19-s − 2.50e68·20-s + 3.69e69·21-s + 4.23e69·22-s − 8.50e70·23-s + 4.12e72·24-s + 7.04e73·25-s − 7.22e73·26-s − 1.54e74·27-s + 5.88e75·28-s + ⋯
L(s)  = 1  + 0.605·2-s − 1.44·3-s − 0.633·4-s + 1.96·5-s − 0.872·6-s − 0.983·7-s − 0.989·8-s + 1.07·9-s + 1.18·10-s + 0.233·11-s + 0.912·12-s − 0.617·13-s − 0.595·14-s − 2.83·15-s + 0.0339·16-s + 1.33·17-s + 0.652·18-s + 0.801·19-s − 1.24·20-s + 1.41·21-s + 0.141·22-s − 0.274·23-s + 1.42·24-s + 2.85·25-s − 0.373·26-s − 0.110·27-s + 0.622·28-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 \( 1 - 3.85e15T + 4.05e31T^{2} \)
3 \( 1 + 1.61e25T + 1.25e50T^{2} \)
5 \( 1 - 9.75e36T + 2.46e73T^{2} \)
7 \( 1 + 2.29e44T + 5.43e88T^{2} \)
11 \( 1 - 1.09e54T + 2.21e109T^{2} \)
13 \( 1 + 1.87e58T + 9.20e116T^{2} \)
17 \( 1 - 5.30e64T + 1.57e129T^{2} \)
19 \( 1 - 1.09e67T + 1.85e134T^{2} \)
23 \( 1 + 8.50e70T + 9.58e142T^{2} \)
29 \( 1 + 5.56e76T + 3.56e153T^{2} \)
31 \( 1 - 1.17e78T + 3.91e156T^{2} \)
37 \( 1 + 2.00e82T + 4.58e164T^{2} \)
41 \( 1 + 1.54e84T + 2.19e169T^{2} \)
43 \( 1 - 5.42e84T + 3.26e171T^{2} \)
47 \( 1 + 1.19e88T + 3.71e175T^{2} \)
53 \( 1 - 1.16e90T + 1.11e181T^{2} \)
59 \( 1 + 1.35e93T + 8.69e185T^{2} \)
61 \( 1 - 9.28e92T + 2.88e187T^{2} \)
67 \( 1 + 5.61e94T + 5.46e191T^{2} \)
71 \( 1 + 1.67e97T + 2.41e194T^{2} \)
73 \( 1 - 8.53e97T + 4.45e195T^{2} \)
79 \( 1 + 4.40e99T + 1.78e199T^{2} \)
83 \( 1 + 2.96e99T + 3.18e201T^{2} \)
89 \( 1 - 2.97e101T + 4.85e204T^{2} \)
97 \( 1 + 1.35e104T + 4.08e208T^{2} \)
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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

−13.14570564705444239315609907885, −12.16332208156089539365401343911, −10.12359189976217410088124973822, −9.484704644244628482653083568705, −6.53366013420896353467939337223, −5.69184239315667961335582970696, −5.05648585599238806439859479947, −3.10325350480788541337629664389, −1.30023151814795063559916376814, 0, 1.30023151814795063559916376814, 3.10325350480788541337629664389, 5.05648585599238806439859479947, 5.69184239315667961335582970696, 6.53366013420896353467939337223, 9.484704644244628482653083568705, 10.12359189976217410088124973822, 12.16332208156089539365401343911, 13.14570564705444239315609907885

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