Properties

Label 2-2e4-16.5-c3-0-1
Degree $2$
Conductor $16$
Sign $0.690 - 0.723i$
Analytic cond. $0.944030$
Root an. cond. $0.971612$
Motivic weight $3$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.690 - 0.723i$
Analytic conductor: \(0.944030\)
Root analytic conductor: \(0.971612\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :3/2),\ 0.690 - 0.723i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.05836 + 0.452995i\)
\(L(\frac12)\) \(\approx\) \(1.05836 + 0.452995i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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