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  + 1.79e13·2-s − 3.11e21·3-s − 2.95e26·4-s − 4.75e30·5-s − 5.59e34·6-s + 3.90e37·7-s − 1.64e40·8-s + 6.77e42·9-s − 8.55e43·10-s − 2.01e45·11-s + 9.19e47·12-s − 1.05e49·13-s + 7.02e50·14-s + 1.48e52·15-s − 1.12e53·16-s + 1.05e55·17-s + 1.21e56·18-s + 1.08e57·19-s + 1.40e57·20-s − 1.21e59·21-s − 3.61e58·22-s − 5.03e60·23-s + 5.11e61·24-s − 1.38e62·25-s − 1.88e62·26-s − 1.20e64·27-s − 1.15e64·28-s + ⋯
L(s)  = 1  + 0.722·2-s − 1.82·3-s − 0.477·4-s − 0.374·5-s − 1.31·6-s + 0.965·7-s − 1.06·8-s + 2.32·9-s − 0.270·10-s − 0.0915·11-s + 0.870·12-s − 0.282·13-s + 0.697·14-s + 0.682·15-s − 0.294·16-s + 1.84·17-s + 1.68·18-s + 1.34·19-s + 0.178·20-s − 1.76·21-s − 0.0661·22-s − 1.27·23-s + 1.94·24-s − 0.859·25-s − 0.204·26-s − 2.42·27-s − 0.460·28-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 \( 1 - 1.79e13T + 6.18e26T^{2} \)
3 \( 1 + 3.11e21T + 2.90e42T^{2} \)
5 \( 1 + 4.75e30T + 1.61e62T^{2} \)
7 \( 1 - 3.90e37T + 1.63e75T^{2} \)
11 \( 1 + 2.01e45T + 4.83e92T^{2} \)
13 \( 1 + 1.05e49T + 1.38e99T^{2} \)
17 \( 1 - 1.05e55T + 3.23e109T^{2} \)
19 \( 1 - 1.08e57T + 6.44e113T^{2} \)
23 \( 1 + 5.03e60T + 1.56e121T^{2} \)
29 \( 1 - 1.12e65T + 1.42e130T^{2} \)
31 \( 1 + 3.18e66T + 5.38e132T^{2} \)
37 \( 1 + 6.99e69T + 3.71e139T^{2} \)
41 \( 1 + 7.38e70T + 3.44e143T^{2} \)
43 \( 1 - 4.97e72T + 2.39e145T^{2} \)
47 \( 1 - 5.91e73T + 6.55e148T^{2} \)
53 \( 1 - 2.39e76T + 2.88e153T^{2} \)
59 \( 1 + 1.72e78T + 4.03e157T^{2} \)
61 \( 1 - 1.90e79T + 7.84e158T^{2} \)
67 \( 1 - 7.04e80T + 3.31e162T^{2} \)
71 \( 1 + 1.27e82T + 5.78e164T^{2} \)
73 \( 1 + 3.75e82T + 6.85e165T^{2} \)
79 \( 1 - 3.21e83T + 7.74e168T^{2} \)
83 \( 1 + 2.22e85T + 6.27e170T^{2} \)
89 \( 1 + 5.54e86T + 3.13e173T^{2} \)
97 \( 1 - 4.88e88T + 6.64e176T^{2} \)
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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

−14.20351999191822931679457585945, −12.30645702256344786830921961182, −11.67916635348701792190761252911, −10.03598843213094312644382738188, −7.62060452023816552695209515151, −5.72401450927163662784671482604, −5.10416049686817180215745340848, −3.84868474186927879830908423001, −1.20485868071045966707975881821, 0, 1.20485868071045966707975881821, 3.84868474186927879830908423001, 5.10416049686817180215745340848, 5.72401450927163662784671482604, 7.62060452023816552695209515151, 10.03598843213094312644382738188, 11.67916635348701792190761252911, 12.30645702256344786830921961182, 14.20351999191822931679457585945

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