Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 89
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.92e13·2-s + 1.59e21·3-s − 2.48e26·4-s − 3.02e30·5-s + 3.07e34·6-s + 1.70e37·7-s − 1.66e40·8-s − 3.61e41·9-s − 5.82e43·10-s + 1.93e46·11-s − 3.96e47·12-s + 1.74e49·13-s + 3.28e50·14-s − 4.83e51·15-s − 1.67e53·16-s − 6.51e54·17-s − 6.96e54·18-s − 9.59e56·19-s + 7.53e56·20-s + 2.72e58·21-s + 3.71e59·22-s − 9.89e59·23-s − 2.66e61·24-s − 1.52e62·25-s + 3.35e62·26-s − 5.22e63·27-s − 4.24e63·28-s + ⋯
L(s)  = 1  + 0.773·2-s + 0.935·3-s − 0.401·4-s − 0.238·5-s + 0.723·6-s + 0.421·7-s − 1.08·8-s − 0.124·9-s − 0.184·10-s + 0.878·11-s − 0.375·12-s + 0.468·13-s + 0.326·14-s − 0.222·15-s − 0.436·16-s − 1.14·17-s − 0.0961·18-s − 1.19·19-s + 0.0957·20-s + 0.394·21-s + 0.679·22-s − 0.250·23-s − 1.01·24-s − 0.943·25-s + 0.362·26-s − 1.05·27-s − 0.169·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(90-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+89/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(89\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1,\ (\ :89/2),\ -1)\)
\(L(45)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{91}{2})\)   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 \( 1 - 1.92e13T + 6.18e26T^{2} \)
3 \( 1 - 1.59e21T + 2.90e42T^{2} \)
5 \( 1 + 3.02e30T + 1.61e62T^{2} \)
7 \( 1 - 1.70e37T + 1.63e75T^{2} \)
11 \( 1 - 1.93e46T + 4.83e92T^{2} \)
13 \( 1 - 1.74e49T + 1.38e99T^{2} \)
17 \( 1 + 6.51e54T + 3.23e109T^{2} \)
19 \( 1 + 9.59e56T + 6.44e113T^{2} \)
23 \( 1 + 9.89e59T + 1.56e121T^{2} \)
29 \( 1 + 1.35e65T + 1.42e130T^{2} \)
31 \( 1 - 1.59e66T + 5.38e132T^{2} \)
37 \( 1 + 2.47e68T + 3.71e139T^{2} \)
41 \( 1 - 4.58e71T + 3.44e143T^{2} \)
43 \( 1 + 7.07e72T + 2.39e145T^{2} \)
47 \( 1 - 1.32e74T + 6.55e148T^{2} \)
53 \( 1 + 9.22e76T + 2.88e153T^{2} \)
59 \( 1 + 8.06e78T + 4.03e157T^{2} \)
61 \( 1 - 4.49e79T + 7.84e158T^{2} \)
67 \( 1 - 1.54e81T + 3.31e162T^{2} \)
71 \( 1 + 2.24e82T + 5.78e164T^{2} \)
73 \( 1 + 1.49e83T + 6.85e165T^{2} \)
79 \( 1 - 3.12e84T + 7.74e168T^{2} \)
83 \( 1 - 3.77e85T + 6.27e170T^{2} \)
89 \( 1 - 2.53e85T + 3.13e173T^{2} \)
97 \( 1 - 3.77e87T + 6.64e176T^{2} \)
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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

−14.25436902683609710519639547749, −13.15768652745913581412522944007, −11.47997200631952938788788758878, −9.216136507943234817028258120466, −8.248137203898072229707984765927, −6.19462237036591940926645435560, −4.44123371708524621083614394267, −3.50641892894935739851514565752, −2.00434077516919755157598282934, 0, 2.00434077516919755157598282934, 3.50641892894935739851514565752, 4.44123371708524621083614394267, 6.19462237036591940926645435560, 8.248137203898072229707984765927, 9.216136507943234817028258120466, 11.47997200631952938788788758878, 13.15768652745913581412522944007, 14.25436902683609710519639547749

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