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Properties

Label 2-2e4-16.13-c13-0-10
Degree $2$
Conductor $16$
Sign $0.186 + 0.982i$
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  + (−30.3 − 85.2i)2-s + (−26.4 − 26.4i)3-s + (−6.35e3 + 5.16e3i)4-s + (−3.80e4 + 3.80e4i)5-s + (−1.45e3 + 3.06e3i)6-s − 2.37e4i·7-s + (6.33e5 + 3.85e5i)8-s − 1.59e6i·9-s + (4.40e6 + 2.09e6i)10-s + (−7.42e5 + 7.42e5i)11-s + (3.05e5 + 3.13e4i)12-s + (7.10e6 + 7.10e6i)13-s + (−2.02e6 + 7.18e5i)14-s + 2.01e6·15-s + (1.36e7 − 6.57e7i)16-s + 1.36e8·17-s + ⋯
L(s)  = 1  + (−0.334 − 0.942i)2-s + (−0.0209 − 0.0209i)3-s + (−0.775 + 0.631i)4-s + (−1.08 + 1.08i)5-s + (−0.0127 + 0.0267i)6-s − 0.0761i·7-s + (0.854 + 0.519i)8-s − 0.999i·9-s + (1.39 + 0.661i)10-s + (−0.126 + 0.126i)11-s + (0.0294 + 0.00303i)12-s + (0.408 + 0.408i)13-s + (−0.0717 + 0.0255i)14-s + 0.0456·15-s + (0.203 − 0.979i)16-s + 1.37·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.186 + 0.982i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.186 + 0.982i$
Analytic conductor: \(17.1569\)
Root analytic conductor: \(4.14209\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :13/2),\ 0.186 + 0.982i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.769975 - 0.637398i\)
\(L(\frac12)\) \(\approx\) \(0.769975 - 0.637398i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (30.3 + 85.2i)T \)
good3 \( 1 + (26.4 + 26.4i)T + 1.59e6iT^{2} \)
5 \( 1 + (3.80e4 - 3.80e4i)T - 1.22e9iT^{2} \)
7 \( 1 + 2.37e4iT - 9.68e10T^{2} \)
11 \( 1 + (7.42e5 - 7.42e5i)T - 3.45e13iT^{2} \)
13 \( 1 + (-7.10e6 - 7.10e6i)T + 3.02e14iT^{2} \)
17 \( 1 - 1.36e8T + 9.90e15T^{2} \)
19 \( 1 + (2.30e8 + 2.30e8i)T + 4.20e16iT^{2} \)
23 \( 1 + 3.78e8iT - 5.04e17T^{2} \)
29 \( 1 + (-1.40e9 - 1.40e9i)T + 1.02e19iT^{2} \)
31 \( 1 - 8.92e9T + 2.44e19T^{2} \)
37 \( 1 + (4.59e9 - 4.59e9i)T - 2.43e20iT^{2} \)
41 \( 1 + 2.11e10iT - 9.25e20T^{2} \)
43 \( 1 + (9.35e9 - 9.35e9i)T - 1.71e21iT^{2} \)
47 \( 1 - 9.81e10T + 5.46e21T^{2} \)
53 \( 1 + (-3.80e10 + 3.80e10i)T - 2.60e22iT^{2} \)
59 \( 1 + (-3.94e11 + 3.94e11i)T - 1.04e23iT^{2} \)
61 \( 1 + (3.39e11 + 3.39e11i)T + 1.61e23iT^{2} \)
67 \( 1 + (-6.51e11 - 6.51e11i)T + 5.48e23iT^{2} \)
71 \( 1 + 4.47e11iT - 1.16e24T^{2} \)
73 \( 1 - 2.00e12iT - 1.67e24T^{2} \)
79 \( 1 + 9.96e11T + 4.66e24T^{2} \)
83 \( 1 + (1.95e12 + 1.95e12i)T + 8.87e24iT^{2} \)
89 \( 1 + 4.25e12iT - 2.19e25T^{2} \)
97 \( 1 + 1.11e13T + 6.73e25T^{2} \)
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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

−15.49960858730928994710537174787, −14.22576158483460384685434857480, −12.39585880529814655210776046540, −11.40545034790701412735861540589, −10.21653713245247712515721712787, −8.487630372291992973544593103190, −6.92013342772174906100593905608, −4.08007437954793146445940499533, −2.88347300391173293170328501664, −0.61362071965793988154310205461, 0.965455929264708982474088531637, 4.22599113106331244797834568131, 5.59767821080658853593555289827, 7.79517689036165365538840785190, 8.449692821897930876197537669230, 10.32344304644935937773101166892, 12.24439594269984029547896364609, 13.65258216896097755056200979283, 15.26697979006534823030279103174, 16.27834372632688770985656646046

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