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Properties

Label 2-2e4-16.5-c7-0-6
Degree $2$
Conductor $16$
Sign $0.250 + 0.968i$
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  + (−11.0 − 2.59i)2-s + (−56.5 + 56.5i)3-s + (114. + 57.0i)4-s + (−90.1 − 90.1i)5-s + (769. − 476. i)6-s − 373. i·7-s + (−1.11e3 − 925. i)8-s − 4.21e3i·9-s + (759. + 1.22e3i)10-s + (2.84e3 + 2.84e3i)11-s + (−9.71e3 + 3.25e3i)12-s + (9.46e3 − 9.46e3i)13-s + (−969. + 4.11e3i)14-s + 1.02e4·15-s + (9.86e3 + 1.30e4i)16-s − 3.38e4·17-s + ⋯
L(s)  = 1  + (−0.973 − 0.229i)2-s + (−1.20 + 1.20i)3-s + (0.895 + 0.445i)4-s + (−0.322 − 0.322i)5-s + (1.45 − 0.900i)6-s − 0.412i·7-s + (−0.769 − 0.639i)8-s − 1.92i·9-s + (0.240 + 0.387i)10-s + (0.644 + 0.644i)11-s + (−1.62 + 0.543i)12-s + (1.19 − 1.19i)13-s + (−0.0943 + 0.401i)14-s + 0.780·15-s + (0.602 + 0.798i)16-s − 1.66·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.250 + 0.968i$
Analytic conductor: \(4.99816\)
Root analytic conductor: \(2.23565\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :7/2),\ 0.250 + 0.968i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.306961 - 0.237566i\)
\(L(\frac12)\) \(\approx\) \(0.306961 - 0.237566i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (11.0 + 2.59i)T \)
good3 \( 1 + (56.5 - 56.5i)T - 2.18e3iT^{2} \)
5 \( 1 + (90.1 + 90.1i)T + 7.81e4iT^{2} \)
7 \( 1 + 373. iT - 8.23e5T^{2} \)
11 \( 1 + (-2.84e3 - 2.84e3i)T + 1.94e7iT^{2} \)
13 \( 1 + (-9.46e3 + 9.46e3i)T - 6.27e7iT^{2} \)
17 \( 1 + 3.38e4T + 4.10e8T^{2} \)
19 \( 1 + (6.16e3 - 6.16e3i)T - 8.93e8iT^{2} \)
23 \( 1 + 8.53e4iT - 3.40e9T^{2} \)
29 \( 1 + (3.15e3 - 3.15e3i)T - 1.72e10iT^{2} \)
31 \( 1 - 7.05e4T + 2.75e10T^{2} \)
37 \( 1 + (1.50e5 + 1.50e5i)T + 9.49e10iT^{2} \)
41 \( 1 + 4.05e5iT - 1.94e11T^{2} \)
43 \( 1 + (-1.39e5 - 1.39e5i)T + 2.71e11iT^{2} \)
47 \( 1 + 6.95e5T + 5.06e11T^{2} \)
53 \( 1 + (-4.51e5 - 4.51e5i)T + 1.17e12iT^{2} \)
59 \( 1 + (1.50e6 + 1.50e6i)T + 2.48e12iT^{2} \)
61 \( 1 + (-3.32e5 + 3.32e5i)T - 3.14e12iT^{2} \)
67 \( 1 + (2.89e5 - 2.89e5i)T - 6.06e12iT^{2} \)
71 \( 1 + 1.04e5iT - 9.09e12T^{2} \)
73 \( 1 + 2.59e6iT - 1.10e13T^{2} \)
79 \( 1 + 6.18e6T + 1.92e13T^{2} \)
83 \( 1 + (1.99e6 - 1.99e6i)T - 2.71e13iT^{2} \)
89 \( 1 - 2.06e6iT - 4.42e13T^{2} \)
97 \( 1 + 4.06e6T + 8.07e13T^{2} \)
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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

−17.23861098469693212903689370260, −16.12817266268264093630178751740, −15.35662130045460661579403639343, −12.44760001664741099426598639515, −11.07832200157892539852955456807, −10.30489779917398220552341735907, −8.728739771339511318302926371980, −6.40143124603047203181055661242, −4.18914988258403496629000389570, −0.40358690853386958280764132251, 1.49654035789573222048653269780, 6.08779038907440635815227185141, 7.00688693673517586175922238301, 8.790882257440019706504343064407, 11.24891187056461223138277524748, 11.55820105977124429194314397491, 13.51582619410939436902011364216, 15.61886040283702626804088710913, 16.84463287142787164737196665611, 17.86966457360294130362637987424

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