Properties

Degree 2
Conductor 2
Sign $1$
Motivic weight 49
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.67e7·2-s − 1.33e11·3-s + 2.81e14·4-s + 1.87e17·5-s − 2.23e18·6-s + 8.28e20·7-s + 4.72e21·8-s − 2.21e23·9-s + 3.14e24·10-s − 3.05e25·11-s − 3.75e25·12-s − 4.92e25·13-s + 1.38e28·14-s − 2.50e28·15-s + 7.92e28·16-s + 2.50e30·17-s − 3.71e30·18-s + 5.26e30·19-s + 5.28e31·20-s − 1.10e32·21-s − 5.12e32·22-s − 7.99e32·23-s − 6.29e32·24-s + 1.74e34·25-s − 8.26e32·26-s + 6.14e34·27-s + 2.33e35·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.272·3-s + 0.5·4-s + 1.40·5-s − 0.192·6-s + 1.63·7-s + 0.353·8-s − 0.925·9-s + 0.996·10-s − 0.935·11-s − 0.136·12-s − 0.0251·13-s + 1.15·14-s − 0.384·15-s + 0.250·16-s + 1.79·17-s − 0.654·18-s + 0.246·19-s + 0.704·20-s − 0.445·21-s − 0.661·22-s − 0.347·23-s − 0.0964·24-s + 0.984·25-s − 0.0178·26-s + 0.525·27-s + 0.817·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(50-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+49/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(49\)
character  :  $\chi_{2} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 2,\ (\ :49/2),\ 1)$
$L(25)$  $\approx$  $4.123845134$
$L(\frac12)$  $\approx$  $4.123845134$
$L(\frac{51}{2})$   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 \( 1 - 1.67e7T \)
good3 \( 1 + 1.33e11T + 2.39e23T^{2} \)
5 \( 1 - 1.87e17T + 1.77e34T^{2} \)
7 \( 1 - 8.28e20T + 2.56e41T^{2} \)
11 \( 1 + 3.05e25T + 1.06e51T^{2} \)
13 \( 1 + 4.92e25T + 3.83e54T^{2} \)
17 \( 1 - 2.50e30T + 1.95e60T^{2} \)
19 \( 1 - 5.26e30T + 4.55e62T^{2} \)
23 \( 1 + 7.99e32T + 5.30e66T^{2} \)
29 \( 1 + 2.91e35T + 4.54e71T^{2} \)
31 \( 1 - 5.98e36T + 1.19e73T^{2} \)
37 \( 1 - 1.31e38T + 6.94e76T^{2} \)
41 \( 1 + 2.57e39T + 1.06e79T^{2} \)
43 \( 1 - 3.01e39T + 1.09e80T^{2} \)
47 \( 1 + 1.35e41T + 8.56e81T^{2} \)
53 \( 1 - 3.38e42T + 3.08e84T^{2} \)
59 \( 1 + 1.39e42T + 5.91e86T^{2} \)
61 \( 1 + 8.43e43T + 3.02e87T^{2} \)
67 \( 1 - 4.28e44T + 3.00e89T^{2} \)
71 \( 1 + 1.65e45T + 5.14e90T^{2} \)
73 \( 1 + 5.25e45T + 2.00e91T^{2} \)
79 \( 1 - 1.45e46T + 9.63e92T^{2} \)
83 \( 1 + 9.24e46T + 1.08e94T^{2} \)
89 \( 1 - 8.22e46T + 3.31e95T^{2} \)
97 \( 1 + 4.76e48T + 2.24e97T^{2} \)
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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

−16.94338572577514840230696520490, −14.63842697430683460631080616061, −13.69632637739654228276171914503, −11.78024926452517289789238985645, −10.28255603028618506949661735125, −8.029601410615503291712199546654, −5.78881220888448067201822684551, −5.05986559254949980047816221721, −2.65988708517971336327308455761, −1.35713215212954564094521439019, 1.35713215212954564094521439019, 2.65988708517971336327308455761, 5.05986559254949980047816221721, 5.78881220888448067201822684551, 8.029601410615503291712199546654, 10.28255603028618506949661735125, 11.78024926452517289789238985645, 13.69632637739654228276171914503, 14.63842697430683460631080616061, 16.94338572577514840230696520490

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