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Properties

Label 2-2e4-16.3-c2-0-0
Degree $2$
Conductor $16$
Sign $0.696 - 0.717i$
Analytic cond. $0.435968$
Root an. cond. $0.660279$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.573 + 1.91i)2-s + (0.146 − 0.146i)3-s + (−3.34 − 2.19i)4-s + (3.68 − 3.68i)5-s + (0.196 + 0.364i)6-s − 9.66·7-s + (6.12 − 5.14i)8-s + 8.95i·9-s + (4.94 + 9.17i)10-s + (5.51 + 5.51i)11-s + (−0.810 + 0.167i)12-s + (−6.27 − 6.27i)13-s + (5.53 − 18.5i)14-s − 1.07i·15-s + (6.35 + 14.6i)16-s − 6.78·17-s + ⋯
L(s)  = 1  + (−0.286 + 0.958i)2-s + (0.0487 − 0.0487i)3-s + (−0.835 − 0.549i)4-s + (0.737 − 0.737i)5-s + (0.0327 + 0.0607i)6-s − 1.38·7-s + (0.765 − 0.643i)8-s + 0.995i·9-s + (0.494 + 0.917i)10-s + (0.501 + 0.501i)11-s + (−0.0675 + 0.0139i)12-s + (−0.482 − 0.482i)13-s + (0.395 − 1.32i)14-s − 0.0719i·15-s + (0.396 + 0.917i)16-s − 0.399·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.696 - 0.717i$
Analytic conductor: \(0.435968\)
Root analytic conductor: \(0.660279\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :1),\ 0.696 - 0.717i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.638050 + 0.270074i\)
\(L(\frac12)\) \(\approx\) \(0.638050 + 0.270074i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.573 - 1.91i)T \)
good3 \( 1 + (-0.146 + 0.146i)T - 9iT^{2} \)
5 \( 1 + (-3.68 + 3.68i)T - 25iT^{2} \)
7 \( 1 + 9.66T + 49T^{2} \)
11 \( 1 + (-5.51 - 5.51i)T + 121iT^{2} \)
13 \( 1 + (6.27 + 6.27i)T + 169iT^{2} \)
17 \( 1 + 6.78T + 289T^{2} \)
19 \( 1 + (-13.5 + 13.5i)T - 361iT^{2} \)
23 \( 1 - 17.0T + 529T^{2} \)
29 \( 1 + (-4.85 - 4.85i)T + 841iT^{2} \)
31 \( 1 + 5.25iT - 961T^{2} \)
37 \( 1 + (18.1 - 18.1i)T - 1.36e3iT^{2} \)
41 \( 1 + 48.2iT - 1.68e3T^{2} \)
43 \( 1 + (54.5 + 54.5i)T + 1.84e3iT^{2} \)
47 \( 1 - 40.4iT - 2.20e3T^{2} \)
53 \( 1 + (-10.8 + 10.8i)T - 2.80e3iT^{2} \)
59 \( 1 + (-50.8 - 50.8i)T + 3.48e3iT^{2} \)
61 \( 1 + (17.0 + 17.0i)T + 3.72e3iT^{2} \)
67 \( 1 + (-22.9 + 22.9i)T - 4.48e3iT^{2} \)
71 \( 1 + 51.6T + 5.04e3T^{2} \)
73 \( 1 - 78.5iT - 5.32e3T^{2} \)
79 \( 1 - 108. iT - 6.24e3T^{2} \)
83 \( 1 + (-57.3 + 57.3i)T - 6.88e3iT^{2} \)
89 \( 1 - 44.1iT - 7.92e3T^{2} \)
97 \( 1 - 112.T + 9.40e3T^{2} \)
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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

−19.05917366499240579387292539661, −17.44809472315373685434368322205, −16.61854860503053858588883402170, −15.48577849509374745221103926577, −13.70565852781556031159673061557, −12.86943297645273560521880912709, −10.06652148021096439654084648890, −8.994231981046499986081599454075, −7.00661769518024865851982438880, −5.23955131995947565398025839100, 3.26337001318846697709200002217, 6.53005724998402159368645936403, 9.230626755154415845242817721063, 10.09800024692954469366961831329, 11.82170795194160262411601729103, 13.20271751375459428613970751030, 14.48541207774191503482784312224, 16.50844412781028768059825791010, 17.85434448226598555919006841803, 18.90176009244069779044019867640

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