Properties

Degree 2
Conductor 2
Sign $1$
Motivic weight 83
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.19e12·2-s + 2.32e18·3-s + 4.83e24·4-s + 1.63e29·5-s − 5.11e30·6-s − 8.77e33·7-s − 1.06e37·8-s − 3.98e39·9-s − 3.59e41·10-s − 4.41e42·11-s + 1.12e43·12-s − 1.03e45·13-s + 1.93e46·14-s + 3.80e47·15-s + 2.33e49·16-s + 7.89e50·17-s + 8.76e51·18-s + 1.66e53·19-s + 7.91e53·20-s − 2.03e52·21-s + 9.70e54·22-s − 3.15e56·23-s − 2.47e55·24-s + 1.64e58·25-s + 2.27e57·26-s − 1.85e58·27-s − 4.24e58·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.0367·3-s + 0.5·4-s + 1.60·5-s − 0.0260·6-s − 0.0744·7-s − 0.353·8-s − 0.998·9-s − 1.13·10-s − 0.267·11-s + 0.0183·12-s − 0.0610·13-s + 0.0526·14-s + 0.0592·15-s + 0.250·16-s + 0.681·17-s + 0.706·18-s + 1.42·19-s + 0.804·20-s − 0.00273·21-s + 0.188·22-s − 0.971·23-s − 0.0130·24-s + 1.58·25-s + 0.0431·26-s − 0.0735·27-s − 0.0372·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(84-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+83/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(83\)
character  :  $\chi_{2} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 2,\ (\ :83/2),\ 1)$
$L(42)$  $\approx$  $2.027545440$
$L(\frac12)$  $\approx$  $2.027545440$
$L(\frac{85}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + 2.19e12T \)
good3 \( 1 - 2.32e18T + 3.99e39T^{2} \)
5 \( 1 - 1.63e29T + 1.03e58T^{2} \)
7 \( 1 + 8.77e33T + 1.39e70T^{2} \)
11 \( 1 + 4.41e42T + 2.72e86T^{2} \)
13 \( 1 + 1.03e45T + 2.86e92T^{2} \)
17 \( 1 - 7.89e50T + 1.34e102T^{2} \)
19 \( 1 - 1.66e53T + 1.36e106T^{2} \)
23 \( 1 + 3.15e56T + 1.05e113T^{2} \)
29 \( 1 + 7.28e60T + 2.39e121T^{2} \)
31 \( 1 - 1.29e62T + 6.06e123T^{2} \)
37 \( 1 - 1.79e65T + 1.44e130T^{2} \)
41 \( 1 - 3.38e66T + 7.26e133T^{2} \)
43 \( 1 + 3.48e67T + 3.78e135T^{2} \)
47 \( 1 + 1.23e69T + 6.08e138T^{2} \)
53 \( 1 - 3.80e71T + 1.30e143T^{2} \)
59 \( 1 - 2.39e73T + 9.56e146T^{2} \)
61 \( 1 + 3.26e73T + 1.52e148T^{2} \)
67 \( 1 + 5.45e75T + 3.66e151T^{2} \)
71 \( 1 - 6.95e76T + 4.51e153T^{2} \)
73 \( 1 - 9.48e76T + 4.52e154T^{2} \)
79 \( 1 + 8.48e78T + 3.18e157T^{2} \)
83 \( 1 - 4.88e79T + 1.92e159T^{2} \)
89 \( 1 - 1.14e81T + 6.30e161T^{2} \)
97 \( 1 - 5.10e82T + 7.98e164T^{2} \)
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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

−13.41379496684283279997256823910, −11.64884179152788444134346151978, −10.10692298276767702176028971077, −9.341850961182419423369906951027, −7.894911232942774342096213026871, −6.20535689093710138932044460470, −5.38990677386099909510799920598, −3.02793283270186350080828190482, −2.01471965848111635953369991381, −0.791014437359105890905061053263, 0.791014437359105890905061053263, 2.01471965848111635953369991381, 3.02793283270186350080828190482, 5.38990677386099909510799920598, 6.20535689093710138932044460470, 7.894911232942774342096213026871, 9.341850961182419423369906951027, 10.10692298276767702176028971077, 11.64884179152788444134346151978, 13.41379496684283279997256823910

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