Properties

Label 2-2e4-16.13-c7-0-4
Degree $2$
Conductor $16$
Sign $-0.358 - 0.933i$
Analytic cond. $4.99816$
Root an. cond. $2.23565$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.26 + 10.4i)2-s + (61.7 + 61.7i)3-s + (−91.7 + 89.3i)4-s + (160. − 160. i)5-s + (−384. + 910. i)6-s − 1.20e3i·7-s + (−1.32e3 − 580. i)8-s + 5.44e3i·9-s + (2.37e3 + 1.00e3i)10-s + (313. − 313. i)11-s + (−1.11e4 − 148. i)12-s + (−2.23e3 − 2.23e3i)13-s + (1.26e4 − 5.12e3i)14-s + 1.98e4·15-s + (434. − 1.63e4i)16-s + 2.10e4·17-s + ⋯
L(s)  = 1  + (0.376 + 0.926i)2-s + (1.32 + 1.32i)3-s + (−0.716 + 0.697i)4-s + (0.575 − 0.575i)5-s + (−0.726 + 1.72i)6-s − 1.32i·7-s + (−0.916 − 0.400i)8-s + 2.48i·9-s + (0.749 + 0.316i)10-s + (0.0710 − 0.0710i)11-s + (−1.86 − 0.0247i)12-s + (−0.281 − 0.281i)13-s + (1.22 − 0.498i)14-s + 1.51·15-s + (0.0265 − 0.999i)16-s + 1.03·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.358 - 0.933i$
Analytic conductor: \(4.99816\)
Root analytic conductor: \(2.23565\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :7/2),\ -0.358 - 0.933i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.45883 + 2.12186i\)
\(L(\frac12)\) \(\approx\) \(1.45883 + 2.12186i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.26 - 10.4i)T \)
good3 \( 1 + (-61.7 - 61.7i)T + 2.18e3iT^{2} \)
5 \( 1 + (-160. + 160. i)T - 7.81e4iT^{2} \)
7 \( 1 + 1.20e3iT - 8.23e5T^{2} \)
11 \( 1 + (-313. + 313. i)T - 1.94e7iT^{2} \)
13 \( 1 + (2.23e3 + 2.23e3i)T + 6.27e7iT^{2} \)
17 \( 1 - 2.10e4T + 4.10e8T^{2} \)
19 \( 1 + (1.79e4 + 1.79e4i)T + 8.93e8iT^{2} \)
23 \( 1 - 3.36e4iT - 3.40e9T^{2} \)
29 \( 1 + (4.10e3 + 4.10e3i)T + 1.72e10iT^{2} \)
31 \( 1 - 9.13e3T + 2.75e10T^{2} \)
37 \( 1 + (1.04e5 - 1.04e5i)T - 9.49e10iT^{2} \)
41 \( 1 + 5.99e5iT - 1.94e11T^{2} \)
43 \( 1 + (-7.98e4 + 7.98e4i)T - 2.71e11iT^{2} \)
47 \( 1 + 1.24e6T + 5.06e11T^{2} \)
53 \( 1 + (-2.74e5 + 2.74e5i)T - 1.17e12iT^{2} \)
59 \( 1 + (1.13e6 - 1.13e6i)T - 2.48e12iT^{2} \)
61 \( 1 + (-9.84e5 - 9.84e5i)T + 3.14e12iT^{2} \)
67 \( 1 + (2.25e6 + 2.25e6i)T + 6.06e12iT^{2} \)
71 \( 1 + 5.48e6iT - 9.09e12T^{2} \)
73 \( 1 - 4.23e6iT - 1.10e13T^{2} \)
79 \( 1 - 3.54e6T + 1.92e13T^{2} \)
83 \( 1 + (-1.79e5 - 1.79e5i)T + 2.71e13iT^{2} \)
89 \( 1 - 2.62e5iT - 4.42e13T^{2} \)
97 \( 1 - 1.53e7T + 8.07e13T^{2} \)
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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

−17.19649924369073850506937748278, −16.42372228017345933200088314342, −15.14669723381694306083590348936, −14.06785424298543779793467545246, −13.26898233141698449802570168797, −10.20961293654961258161216332437, −9.072082234706373274656650644971, −7.67821309537210866540533765070, −4.96747033531760587089910732348, −3.59182186076880941978613677642, 1.85619574025655825543259973628, 2.93161109122682251993242279838, 6.22264173081858712656331581613, 8.400018618360664707196738831071, 9.698789309011735203013562133126, 12.01323367579451210750620349605, 12.89164673083565362239735988867, 14.27830192461827146648517015386, 14.80744295576141719960758292526, 18.02241033786382067558892906669

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