Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 101
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.76e15·2-s − 3.81e23·3-s + 5.85e29·4-s − 3.44e35·5-s + 6.74e38·6-s − 5.35e42·7-s + 3.44e45·8-s − 1.40e48·9-s + 6.07e50·10-s − 3.44e52·11-s − 2.23e53·12-s + 3.09e56·13-s + 9.46e57·14-s + 1.31e59·15-s − 7.56e60·16-s − 1.06e62·17-s + 2.47e63·18-s − 6.11e63·19-s − 2.01e65·20-s + 2.04e66·21-s + 6.08e67·22-s − 3.47e68·23-s − 1.31e69·24-s + 7.89e70·25-s − 5.47e71·26-s + 1.12e72·27-s − 3.13e72·28-s + ⋯
L(s)  = 1  − 1.10·2-s − 0.307·3-s + 0.231·4-s − 1.73·5-s + 0.340·6-s − 1.12·7-s + 0.853·8-s − 0.905·9-s + 1.92·10-s − 0.884·11-s − 0.0709·12-s + 1.72·13-s + 1.24·14-s + 0.531·15-s − 1.17·16-s − 0.773·17-s + 1.00·18-s − 0.161·19-s − 0.400·20-s + 0.345·21-s + 0.981·22-s − 0.593·23-s − 0.261·24-s + 2.00·25-s − 1.91·26-s + 0.585·27-s − 0.260·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(102-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+50.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(101\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1,\ (\ :101/2),\ -1)\)
\(L(51)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{103}{2})\)   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 \( 1 + 1.76e15T + 2.53e30T^{2} \)
3 \( 1 + 3.81e23T + 1.54e48T^{2} \)
5 \( 1 + 3.44e35T + 3.94e70T^{2} \)
7 \( 1 + 5.35e42T + 2.26e85T^{2} \)
11 \( 1 + 3.44e52T + 1.51e105T^{2} \)
13 \( 1 - 3.09e56T + 3.22e112T^{2} \)
17 \( 1 + 1.06e62T + 1.88e124T^{2} \)
19 \( 1 + 6.11e63T + 1.42e129T^{2} \)
23 \( 1 + 3.47e68T + 3.42e137T^{2} \)
29 \( 1 - 5.28e73T + 5.03e147T^{2} \)
31 \( 1 - 2.84e75T + 4.24e150T^{2} \)
37 \( 1 - 1.90e79T + 2.44e158T^{2} \)
41 \( 1 - 6.25e80T + 7.78e162T^{2} \)
43 \( 1 + 1.65e82T + 9.55e164T^{2} \)
47 \( 1 + 1.59e84T + 7.61e168T^{2} \)
53 \( 1 - 9.26e86T + 1.41e174T^{2} \)
59 \( 1 + 2.36e89T + 7.17e178T^{2} \)
61 \( 1 - 8.85e89T + 2.08e180T^{2} \)
67 \( 1 + 3.21e92T + 2.71e184T^{2} \)
71 \( 1 - 1.94e93T + 9.48e186T^{2} \)
73 \( 1 + 1.32e94T + 1.56e188T^{2} \)
79 \( 1 + 3.38e95T + 4.57e191T^{2} \)
83 \( 1 + 8.23e95T + 6.71e193T^{2} \)
89 \( 1 - 2.68e98T + 7.73e196T^{2} \)
97 \( 1 - 1.33e100T + 4.61e200T^{2} \)
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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

−13.29076983535839515507813011148, −11.57166014266276124846262822142, −10.52349476603585916284955486504, −8.723495116881691523653683175258, −7.971941449289832658130180355403, −6.39805000746559468826497672440, −4.33189322316525769573113554062, −3.03400786597146406168646967019, −0.73200119107837448297302939233, 0, 0.73200119107837448297302939233, 3.03400786597146406168646967019, 4.33189322316525769573113554062, 6.39805000746559468826497672440, 7.971941449289832658130180355403, 8.723495116881691523653683175258, 10.52349476603585916284955486504, 11.57166014266276124846262822142, 13.29076983535839515507813011148

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