Properties

Degree 2
Conductor $ 2^{4} $
Sign $0.690 + 0.723i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.836 − 2.70i)2-s + (1.98 + 1.98i)3-s + (−6.59 − 4.52i)4-s + (−0.596 + 0.596i)5-s + (7.01 − 3.69i)6-s + 29.0i·7-s + (−17.7 + 14.0i)8-s − 19.1i·9-s + (1.11 + 2.11i)10-s + (12.1 − 12.1i)11-s + (−4.11 − 22.0i)12-s + (−48.5 − 48.5i)13-s + (78.5 + 24.3i)14-s − 2.36·15-s + (23.0 + 59.6i)16-s + 86.7·17-s + ⋯
L(s)  = 1  + (0.295 − 0.955i)2-s + (0.381 + 0.381i)3-s + (−0.824 − 0.565i)4-s + (−0.0533 + 0.0533i)5-s + (0.477 − 0.251i)6-s + 1.57i·7-s + (−0.784 + 0.620i)8-s − 0.708i·9-s + (0.0351 + 0.0667i)10-s + (0.332 − 0.332i)11-s + (−0.0991 − 0.530i)12-s + (−1.03 − 1.03i)13-s + (1.50 + 0.464i)14-s − 0.0407·15-s + (0.360 + 0.932i)16-s + 1.23·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $0.690 + 0.723i$
motivic weight  =  \(3\)
character  :  $\chi_{16} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :3/2),\ 0.690 + 0.723i)$
$L(2)$  $\approx$  $1.05836 - 0.452995i$
$L(\frac12)$  $\approx$  $1.05836 - 0.452995i$
$L(\frac{5}{2})$   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 \( 1 + (-0.836 + 2.70i)T \)
good3 \( 1 + (-1.98 - 1.98i)T + 27iT^{2} \)
5 \( 1 + (0.596 - 0.596i)T - 125iT^{2} \)
7 \( 1 - 29.0iT - 343T^{2} \)
11 \( 1 + (-12.1 + 12.1i)T - 1.33e3iT^{2} \)
13 \( 1 + (48.5 + 48.5i)T + 2.19e3iT^{2} \)
17 \( 1 - 86.7T + 4.91e3T^{2} \)
19 \( 1 + (54.8 + 54.8i)T + 6.85e3iT^{2} \)
23 \( 1 - 70.2iT - 1.21e4T^{2} \)
29 \( 1 + (-63.4 - 63.4i)T + 2.43e4iT^{2} \)
31 \( 1 + 8.86T + 2.97e4T^{2} \)
37 \( 1 + (21.7 - 21.7i)T - 5.06e4iT^{2} \)
41 \( 1 + 153. iT - 6.89e4T^{2} \)
43 \( 1 + (120. - 120. i)T - 7.95e4iT^{2} \)
47 \( 1 + 99.9T + 1.03e5T^{2} \)
53 \( 1 + (-389. + 389. i)T - 1.48e5iT^{2} \)
59 \( 1 + (324. - 324. i)T - 2.05e5iT^{2} \)
61 \( 1 + (0.339 + 0.339i)T + 2.26e5iT^{2} \)
67 \( 1 + (-565. - 565. i)T + 3.00e5iT^{2} \)
71 \( 1 + 419. iT - 3.57e5T^{2} \)
73 \( 1 - 374. iT - 3.89e5T^{2} \)
79 \( 1 + 705.T + 4.93e5T^{2} \)
83 \( 1 + (947. + 947. i)T + 5.71e5iT^{2} \)
89 \( 1 + 4.72iT - 7.04e5T^{2} \)
97 \( 1 - 379.T + 9.12e5T^{2} \)
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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

−18.81499708562010550193179178208, −17.58272052045953503232993107530, −15.30440907487681536633485048095, −14.62756423135920295629486760238, −12.70323518409211395850602867151, −11.73893113873026473213007782786, −9.874240689495280735030158463244, −8.742826986623313927493260323542, −5.47840845349865906440710770840, −3.06015101561518229796816170911, 4.40303099004854676081851310537, 6.92156721859609382299166795991, 8.013605696082205649768765020321, 10.05390139051174890333001209254, 12.41873289532976287133991246237, 13.88395681024504428049634726154, 14.47141120147056739317482069840, 16.53554221390283811333753072371, 17.02127008544090049789418520416, 18.78089501296863540170105465116

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