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Properties

Label 2-2e4-16.5-c3-0-3
Degree $2$
Conductor $16$
Sign $0.998 - 0.0551i$
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  + (2.81 − 0.316i)2-s + (−3.27 + 3.27i)3-s + (7.80 − 1.77i)4-s + (−12.6 − 12.6i)5-s + (−8.16 + 10.2i)6-s + 13.8i·7-s + (21.3 − 7.45i)8-s + 5.59i·9-s + (−39.5 − 31.5i)10-s + (1.54 + 1.54i)11-s + (−19.7 + 31.3i)12-s + (32.7 − 32.7i)13-s + (4.38 + 38.9i)14-s + 82.7·15-s + (57.6 − 27.7i)16-s + 18.6·17-s + ⋯
L(s)  = 1  + (0.993 − 0.111i)2-s + (−0.629 + 0.629i)3-s + (0.975 − 0.222i)4-s + (−1.13 − 1.13i)5-s + (−0.555 + 0.695i)6-s + 0.749i·7-s + (0.944 − 0.329i)8-s + 0.207i·9-s + (−1.25 − 0.997i)10-s + (0.0424 + 0.0424i)11-s + (−0.474 + 0.753i)12-s + (0.699 − 0.699i)13-s + (0.0837 + 0.744i)14-s + 1.42·15-s + (0.901 − 0.433i)16-s + 0.266·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0551i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.998 - 0.0551i$
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: \((2,\ 16,\ (\ :3/2),\ 0.998 - 0.0551i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.29158 + 0.0356701i\)
\(L(\frac12)\) \(\approx\) \(1.29158 + 0.0356701i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.81 + 0.316i)T \)
good3 \( 1 + (3.27 - 3.27i)T - 27iT^{2} \)
5 \( 1 + (12.6 + 12.6i)T + 125iT^{2} \)
7 \( 1 - 13.8iT - 343T^{2} \)
11 \( 1 + (-1.54 - 1.54i)T + 1.33e3iT^{2} \)
13 \( 1 + (-32.7 + 32.7i)T - 2.19e3iT^{2} \)
17 \( 1 - 18.6T + 4.91e3T^{2} \)
19 \( 1 + (86.4 - 86.4i)T - 6.85e3iT^{2} \)
23 \( 1 + 134. iT - 1.21e4T^{2} \)
29 \( 1 + (59.7 - 59.7i)T - 2.43e4iT^{2} \)
31 \( 1 + 31.5T + 2.97e4T^{2} \)
37 \( 1 + (-89.1 - 89.1i)T + 5.06e4iT^{2} \)
41 \( 1 + 210. iT - 6.89e4T^{2} \)
43 \( 1 + (-119. - 119. i)T + 7.95e4iT^{2} \)
47 \( 1 + 182.T + 1.03e5T^{2} \)
53 \( 1 + (26.1 + 26.1i)T + 1.48e5iT^{2} \)
59 \( 1 + (-441. - 441. i)T + 2.05e5iT^{2} \)
61 \( 1 + (174. - 174. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-91.7 + 91.7i)T - 3.00e5iT^{2} \)
71 \( 1 - 348. iT - 3.57e5T^{2} \)
73 \( 1 + 299. iT - 3.89e5T^{2} \)
79 \( 1 + 943.T + 4.93e5T^{2} \)
83 \( 1 + (-313. + 313. i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.41e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.51e3T + 9.12e5T^{2} \)
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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

−19.01838602946845692153318283135, −16.67359888057166418265649385568, −16.01113049375911288393669519023, −14.96433181711139586642192676086, −12.87078476741698750657766775968, −11.96202222694314605717937679653, −10.67481878802626459797891779872, −8.248769012487150467291391348964, −5.62036500732862361628891850858, −4.23731031807583349022840487675, 3.81746618896785242225556130638, 6.47822961820532445296571330633, 7.43192695854085614181681259807, 11.00004024031805071090457338786, 11.65697398613831378058459120435, 13.20205984725613540839392196906, 14.62429414166078430699001073400, 15.75319592822826627784935850727, 17.24409571198555303481118615532, 18.80428636821950363259418671615

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