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Properties

Label 2-2e4-16.5-c13-0-9
Degree $2$
Conductor $16$
Sign $0.126 - 0.991i$
Analytic cond. $17.1569$
Root an. cond. $4.14209$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−85.7 − 29.0i)2-s + (−924. + 924. i)3-s + (6.50e3 + 4.97e3i)4-s + (4.20e4 + 4.20e4i)5-s + (1.06e5 − 5.24e4i)6-s − 2.91e5i·7-s + (−4.13e5 − 6.15e5i)8-s − 1.14e5i·9-s + (−2.38e6 − 4.82e6i)10-s + (4.32e6 + 4.32e6i)11-s + (−1.06e7 + 1.41e6i)12-s + (1.83e7 − 1.83e7i)13-s + (−8.45e6 + 2.49e7i)14-s − 7.76e7·15-s + (1.76e7 + 6.47e7i)16-s + 1.06e8·17-s + ⋯
L(s)  = 1  + (−0.947 − 0.320i)2-s + (−0.732 + 0.732i)3-s + (0.794 + 0.607i)4-s + (1.20 + 1.20i)5-s + (0.928 − 0.458i)6-s − 0.936i·7-s + (−0.558 − 0.829i)8-s − 0.0719i·9-s + (−0.753 − 1.52i)10-s + (0.736 + 0.736i)11-s + (−1.02 + 0.137i)12-s + (1.05 − 1.05i)13-s + (−0.300 + 0.886i)14-s − 1.76·15-s + (0.262 + 0.964i)16-s + 1.06·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.126 - 0.991i$
Analytic conductor: \(17.1569\)
Root analytic conductor: \(4.14209\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :13/2),\ 0.126 - 0.991i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.931304 + 0.820149i\)
\(L(\frac12)\) \(\approx\) \(0.931304 + 0.820149i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (85.7 + 29.0i)T \)
good3 \( 1 + (924. - 924. i)T - 1.59e6iT^{2} \)
5 \( 1 + (-4.20e4 - 4.20e4i)T + 1.22e9iT^{2} \)
7 \( 1 + 2.91e5iT - 9.68e10T^{2} \)
11 \( 1 + (-4.32e6 - 4.32e6i)T + 3.45e13iT^{2} \)
13 \( 1 + (-1.83e7 + 1.83e7i)T - 3.02e14iT^{2} \)
17 \( 1 - 1.06e8T + 9.90e15T^{2} \)
19 \( 1 + (-5.47e7 + 5.47e7i)T - 4.20e16iT^{2} \)
23 \( 1 - 1.00e9iT - 5.04e17T^{2} \)
29 \( 1 + (1.19e9 - 1.19e9i)T - 1.02e19iT^{2} \)
31 \( 1 + 1.72e9T + 2.44e19T^{2} \)
37 \( 1 + (-1.50e10 - 1.50e10i)T + 2.43e20iT^{2} \)
41 \( 1 - 1.44e10iT - 9.25e20T^{2} \)
43 \( 1 + (3.22e10 + 3.22e10i)T + 1.71e21iT^{2} \)
47 \( 1 + 2.71e10T + 5.46e21T^{2} \)
53 \( 1 + (6.71e9 + 6.71e9i)T + 2.60e22iT^{2} \)
59 \( 1 + (-1.48e11 - 1.48e11i)T + 1.04e23iT^{2} \)
61 \( 1 + (5.33e11 - 5.33e11i)T - 1.61e23iT^{2} \)
67 \( 1 + (-4.15e11 + 4.15e11i)T - 5.48e23iT^{2} \)
71 \( 1 + 1.42e12iT - 1.16e24T^{2} \)
73 \( 1 - 9.89e11iT - 1.67e24T^{2} \)
79 \( 1 + 2.98e12T + 4.66e24T^{2} \)
83 \( 1 + (2.36e10 - 2.36e10i)T - 8.87e24iT^{2} \)
89 \( 1 + 3.85e12iT - 2.19e25T^{2} \)
97 \( 1 + 8.08e12T + 6.73e25T^{2} \)
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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

−16.70123503881204476836965422872, −15.15438526271229980212244151089, −13.49159655687328081999439622967, −11.34157654325887742674101943686, −10.37546515087734616482683884762, −9.761348564907910389530368999971, −7.35369471276611689154954977079, −5.90765384879031127928263927610, −3.38379130989623659973566528751, −1.38338457464302910984074104317, 0.823100001587544492317176231713, 1.74361178124020773313149358157, 5.68590966180486725918306957735, 6.31698684206480321203959602987, 8.596319643688910768745964761388, 9.463668956156499872165832738710, 11.47992218905289936598964737939, 12.58592752471958997853958746390, 14.24858619264340452489788379137, 16.36073611342354274099454156397

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