Properties

Degree 2
Conductor 2
Sign $1$
Motivic weight 43
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.09e6·2-s + 1.06e8·3-s + 4.39e12·4-s − 2.06e15·5-s − 2.23e14·6-s + 1.84e18·7-s − 9.22e18·8-s − 3.28e20·9-s + 4.34e21·10-s − 3.48e22·11-s + 4.68e20·12-s − 4.35e23·13-s − 3.87e24·14-s − 2.20e23·15-s + 1.93e25·16-s − 5.00e24·17-s + 6.88e26·18-s + 2.39e27·19-s − 9.10e27·20-s + 1.96e26·21-s + 7.31e28·22-s + 3.90e27·23-s − 9.82e26·24-s + 3.14e30·25-s + 9.13e29·26-s − 6.99e28·27-s + 8.11e30·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.00588·3-s + 0.5·4-s − 1.94·5-s − 0.00415·6-s + 1.24·7-s − 0.353·8-s − 0.999·9-s + 1.37·10-s − 1.42·11-s + 0.00294·12-s − 0.489·13-s − 0.883·14-s − 0.0114·15-s + 0.250·16-s − 0.0175·17-s + 0.707·18-s + 0.770·19-s − 0.970·20-s + 0.00734·21-s + 1.00·22-s + 0.0206·23-s − 0.00207·24-s + 2.76·25-s + 0.345·26-s − 0.0117·27-s + 0.624·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(44-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(43\)
character  :  $\chi_{2} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 2,\ (\ :43/2),\ 1)$
$L(22)$  $\approx$  $0.658621$
$L(\frac12)$  $\approx$  $0.658621$
$L(\frac{45}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + 2.09e6T \)
good3 \( 1 - 1.06e8T + 3.28e20T^{2} \)
5 \( 1 + 2.06e15T + 1.13e30T^{2} \)
7 \( 1 - 1.84e18T + 2.18e36T^{2} \)
11 \( 1 + 3.48e22T + 6.02e44T^{2} \)
13 \( 1 + 4.35e23T + 7.93e47T^{2} \)
17 \( 1 + 5.00e24T + 8.11e52T^{2} \)
19 \( 1 - 2.39e27T + 9.69e54T^{2} \)
23 \( 1 - 3.90e27T + 3.58e58T^{2} \)
29 \( 1 - 3.01e31T + 7.64e62T^{2} \)
31 \( 1 - 4.83e31T + 1.34e64T^{2} \)
37 \( 1 - 3.30e33T + 2.70e67T^{2} \)
41 \( 1 - 4.18e34T + 2.23e69T^{2} \)
43 \( 1 - 3.60e34T + 1.73e70T^{2} \)
47 \( 1 + 1.27e36T + 7.94e71T^{2} \)
53 \( 1 + 1.39e36T + 1.39e74T^{2} \)
59 \( 1 - 1.27e38T + 1.40e76T^{2} \)
61 \( 1 + 1.47e38T + 5.87e76T^{2} \)
67 \( 1 + 2.49e39T + 3.32e78T^{2} \)
71 \( 1 - 2.98e39T + 4.01e79T^{2} \)
73 \( 1 - 3.76e38T + 1.32e80T^{2} \)
79 \( 1 + 3.92e40T + 3.96e81T^{2} \)
83 \( 1 - 3.07e41T + 3.31e82T^{2} \)
89 \( 1 - 8.59e41T + 6.66e83T^{2} \)
97 \( 1 + 5.70e42T + 2.69e85T^{2} \)
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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

−17.96160645448336421532711381868, −16.09370194834217472692218438686, −14.80832250795167993497721456437, −11.90841038740923063955197932070, −10.94928811191908143949691785540, −8.295072631434559773077479830200, −7.65723300750753107664232097309, −4.86185530703986246145813127644, −2.86481725201026409060499933236, −0.58129219454947557179969273093, 0.58129219454947557179969273093, 2.86481725201026409060499933236, 4.86185530703986246145813127644, 7.65723300750753107664232097309, 8.295072631434559773077479830200, 10.94928811191908143949691785540, 11.90841038740923063955197932070, 14.80832250795167993497721456437, 16.09370194834217472692218438686, 17.96160645448336421532711381868

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