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  + 4.42e13·2-s − 3.38e20·3-s + 1.33e27·4-s + 3.67e30·5-s − 1.49e34·6-s − 4.35e37·7-s + 3.17e40·8-s − 2.79e42·9-s + 1.62e44·10-s − 2.45e46·11-s − 4.53e47·12-s − 6.70e49·13-s − 1.92e51·14-s − 1.24e51·15-s + 5.76e53·16-s + 4.97e54·17-s − 1.23e56·18-s + 4.54e56·19-s + 4.90e57·20-s + 1.47e58·21-s − 1.08e60·22-s + 2.88e60·23-s − 1.07e61·24-s − 1.48e62·25-s − 2.96e63·26-s + 1.93e63·27-s − 5.81e64·28-s + ⋯
L(s)  = 1  + 1.77·2-s − 0.198·3-s + 2.15·4-s + 0.288·5-s − 0.353·6-s − 1.07·7-s + 2.06·8-s − 0.960·9-s + 0.513·10-s − 1.11·11-s − 0.429·12-s − 1.80·13-s − 1.91·14-s − 0.0573·15-s + 1.50·16-s + 0.874·17-s − 1.70·18-s + 0.566·19-s + 0.623·20-s + 0.213·21-s − 1.98·22-s + 0.729·23-s − 0.409·24-s − 0.916·25-s − 3.20·26-s + 0.389·27-s − 2.32·28-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 \( 1 - 4.42e13T + 6.18e26T^{2} \)
3 \( 1 + 3.38e20T + 2.90e42T^{2} \)
5 \( 1 - 3.67e30T + 1.61e62T^{2} \)
7 \( 1 + 4.35e37T + 1.63e75T^{2} \)
11 \( 1 + 2.45e46T + 4.83e92T^{2} \)
13 \( 1 + 6.70e49T + 1.38e99T^{2} \)
17 \( 1 - 4.97e54T + 3.23e109T^{2} \)
19 \( 1 - 4.54e56T + 6.44e113T^{2} \)
23 \( 1 - 2.88e60T + 1.56e121T^{2} \)
29 \( 1 + 1.57e65T + 1.42e130T^{2} \)
31 \( 1 - 4.07e66T + 5.38e132T^{2} \)
37 \( 1 + 7.75e69T + 3.71e139T^{2} \)
41 \( 1 - 2.95e71T + 3.44e143T^{2} \)
43 \( 1 + 1.81e72T + 2.39e145T^{2} \)
47 \( 1 - 1.54e73T + 6.55e148T^{2} \)
53 \( 1 - 3.79e76T + 2.88e153T^{2} \)
59 \( 1 - 4.03e78T + 4.03e157T^{2} \)
61 \( 1 + 2.95e79T + 7.84e158T^{2} \)
67 \( 1 - 5.44e80T + 3.31e162T^{2} \)
71 \( 1 - 6.75e81T + 5.78e164T^{2} \)
73 \( 1 + 1.55e82T + 6.85e165T^{2} \)
79 \( 1 + 4.89e84T + 7.74e168T^{2} \)
83 \( 1 + 5.34e84T + 6.27e170T^{2} \)
89 \( 1 + 7.65e86T + 3.13e173T^{2} \)
97 \( 1 + 1.46e87T + 6.64e176T^{2} \)
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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

−14.10515438874160174513829835801, −12.88070008572923898826358190613, −11.80431143181107961930855792874, −10.01657666711676719925392684605, −7.31415608912075048300202644803, −5.83807579188317149847073664613, −5.03128012492627329283061143280, −3.22121415116759283032440267067, −2.46657337848958541233786904618, 0, 2.46657337848958541233786904618, 3.22121415116759283032440267067, 5.03128012492627329283061143280, 5.83807579188317149847073664613, 7.31415608912075048300202644803, 10.01657666711676719925392684605, 11.80431143181107961930855792874, 12.88070008572923898826358190613, 14.10515438874160174513829835801

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