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Properties

Label 2-2e4-16.13-c3-0-1
Degree $2$
Conductor $16$
Sign $0.499 - 0.866i$
Analytic cond. $0.944030$
Root an. cond. $0.971612$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.460 + 2.79i)2-s + (0.756 + 0.756i)3-s + (−7.57 + 2.57i)4-s + (8.22 − 8.22i)5-s + (−1.76 + 2.46i)6-s − 2.67i·7-s + (−10.6 − 19.9i)8-s − 25.8i·9-s + (26.7 + 19.1i)10-s + (−45.2 + 45.2i)11-s + (−7.67 − 3.78i)12-s + (35.3 + 35.3i)13-s + (7.45 − 1.23i)14-s + 12.4·15-s + (50.7 − 38.9i)16-s − 72.4·17-s + ⋯
L(s)  = 1  + (0.162 + 0.986i)2-s + (0.145 + 0.145i)3-s + (−0.946 + 0.321i)4-s + (0.735 − 0.735i)5-s + (−0.119 + 0.167i)6-s − 0.144i·7-s + (−0.471 − 0.881i)8-s − 0.957i·9-s + (0.845 + 0.605i)10-s + (−1.23 + 1.23i)11-s + (−0.184 − 0.0910i)12-s + (0.755 + 0.755i)13-s + (0.142 − 0.0235i)14-s + 0.214·15-s + (0.793 − 0.609i)16-s − 1.03·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 - 0.866i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.499 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.499 - 0.866i$
Analytic conductor: \(0.944030\)
Root analytic conductor: \(0.971612\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :3/2),\ 0.499 - 0.866i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.943664 + 0.545046i\)
\(L(\frac12)\) \(\approx\) \(0.943664 + 0.545046i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.460 - 2.79i)T \)
good3 \( 1 + (-0.756 - 0.756i)T + 27iT^{2} \)
5 \( 1 + (-8.22 + 8.22i)T - 125iT^{2} \)
7 \( 1 + 2.67iT - 343T^{2} \)
11 \( 1 + (45.2 - 45.2i)T - 1.33e3iT^{2} \)
13 \( 1 + (-35.3 - 35.3i)T + 2.19e3iT^{2} \)
17 \( 1 + 72.4T + 4.91e3T^{2} \)
19 \( 1 + (-19.4 - 19.4i)T + 6.85e3iT^{2} \)
23 \( 1 + 139. iT - 1.21e4T^{2} \)
29 \( 1 + (-66.0 - 66.0i)T + 2.43e4iT^{2} \)
31 \( 1 - 188.T + 2.97e4T^{2} \)
37 \( 1 + (84.0 - 84.0i)T - 5.06e4iT^{2} \)
41 \( 1 - 104. iT - 6.89e4T^{2} \)
43 \( 1 + (31.4 - 31.4i)T - 7.95e4iT^{2} \)
47 \( 1 + 488.T + 1.03e5T^{2} \)
53 \( 1 + (-149. + 149. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-284. + 284. i)T - 2.05e5iT^{2} \)
61 \( 1 + (228. + 228. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-139. - 139. i)T + 3.00e5iT^{2} \)
71 \( 1 + 453. iT - 3.57e5T^{2} \)
73 \( 1 + 259. iT - 3.89e5T^{2} \)
79 \( 1 - 323.T + 4.93e5T^{2} \)
83 \( 1 + (563. + 563. i)T + 5.71e5iT^{2} \)
89 \( 1 - 866. iT - 7.04e5T^{2} \)
97 \( 1 + 936.T + 9.12e5T^{2} \)
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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

−18.30966085760113228255287537381, −17.52471607872774941646223991895, −16.17017258709217705895260223014, −15.06248374774745871847382616081, −13.59133842211992691008963412539, −12.54250575719246046910133861827, −9.848809066563963705704552006510, −8.599453103531040995629116943758, −6.58253413057200100532441752179, −4.70959041005156714992572015765, 2.72333088463365132349406283853, 5.60259368821300895672858813444, 8.357650192784133268869138289171, 10.30085995814503385035996984795, 11.15711219352142456013806911767, 13.35846570225670564111675053452, 13.72613266147089125675496281328, 15.65799274584242342695014859645, 17.72158127192754193792343281251, 18.58524497962152426945672299033

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