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 − 6.77e11·3-s + 2.81e14·4-s − 2.33e17·5-s − 1.13e19·6-s − 5.15e20·7-s + 4.72e21·8-s + 2.19e23·9-s − 3.90e24·10-s − 5.21e25·11-s − 1.90e26·12-s + 1.37e27·13-s − 8.64e27·14-s + 1.57e29·15-s + 7.92e28·16-s − 9.93e29·17-s + 3.67e30·18-s + 1.85e31·19-s − 6.55e31·20-s + 3.49e32·21-s − 8.74e32·22-s − 2.69e33·23-s − 3.19e33·24-s + 3.65e34·25-s + 2.30e34·26-s + 1.35e34·27-s − 1.45e35·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.38·3-s + 0.5·4-s − 1.74·5-s − 0.978·6-s − 1.01·7-s + 0.353·8-s + 0.916·9-s − 1.23·10-s − 1.59·11-s − 0.692·12-s + 0.702·13-s − 0.719·14-s + 2.42·15-s + 0.250·16-s − 0.709·17-s + 0.647·18-s + 0.868·19-s − 0.874·20-s + 1.40·21-s − 1.12·22-s − 1.16·23-s − 0.489·24-s + 2.05·25-s + 0.497·26-s + 0.116·27-s − 0.508·28-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  =  $(2,\ 2,\ (\ :49/2),\ 1)$
$L(25)$  $\approx$  $0.4296860814$
$L(\frac12)$  $\approx$  $0.4296860814$
$L(\frac{51}{2})$   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 \( 1 + 6.77e11T + 2.39e23T^{2} \)
5 \( 1 + 2.33e17T + 1.77e34T^{2} \)
7 \( 1 + 5.15e20T + 2.56e41T^{2} \)
11 \( 1 + 5.21e25T + 1.06e51T^{2} \)
13 \( 1 - 1.37e27T + 3.83e54T^{2} \)
17 \( 1 + 9.93e29T + 1.95e60T^{2} \)
19 \( 1 - 1.85e31T + 4.55e62T^{2} \)
23 \( 1 + 2.69e33T + 5.30e66T^{2} \)
29 \( 1 + 3.88e34T + 4.54e71T^{2} \)
31 \( 1 + 8.58e35T + 1.19e73T^{2} \)
37 \( 1 - 2.50e38T + 6.94e76T^{2} \)
41 \( 1 - 2.12e38T + 1.06e79T^{2} \)
43 \( 1 + 3.48e39T + 1.09e80T^{2} \)
47 \( 1 - 2.11e40T + 8.56e81T^{2} \)
53 \( 1 + 2.52e42T + 3.08e84T^{2} \)
59 \( 1 + 4.29e43T + 5.91e86T^{2} \)
61 \( 1 + 1.42e43T + 3.02e87T^{2} \)
67 \( 1 - 2.95e44T + 3.00e89T^{2} \)
71 \( 1 + 2.56e45T + 5.14e90T^{2} \)
73 \( 1 + 5.17e45T + 2.00e91T^{2} \)
79 \( 1 - 1.28e46T + 9.63e92T^{2} \)
83 \( 1 - 8.58e46T + 1.08e94T^{2} \)
89 \( 1 - 6.95e47T + 3.31e95T^{2} \)
97 \( 1 - 7.53e48T + 2.24e97T^{2} \)
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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

−16.10783579996205814401263107504, −15.76584581557559534952040755148, −12.94587927038859581927827316841, −11.78308669141420651321467337635, −10.71265707610207407597099757239, −7.66940722106239554714971244108, −6.14815329312843700095546581612, −4.66930669446981506766037510014, −3.24898992284991690458622643024, −0.37534543783491370376268612734, 0.37534543783491370376268612734, 3.24898992284991690458622643024, 4.66930669446981506766037510014, 6.14815329312843700095546581612, 7.66940722106239554714971244108, 10.71265707610207407597099757239, 11.78308669141420651321467337635, 12.94587927038859581927827316841, 15.76584581557559534952040755148, 16.10783579996205814401263107504

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