Properties

Label 2-2e4-16.11-c2-0-1
Degree $2$
Conductor $16$
Sign $0.911 - 0.412i$
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  + (1.12 + 1.65i)2-s + (−3.24 − 3.24i)3-s + (−1.47 + 3.71i)4-s + (−0.0586 − 0.0586i)5-s + (1.71 − 9.02i)6-s + 4.61·7-s + (−7.80 + 1.75i)8-s + 12.1i·9-s + (0.0310 − 0.162i)10-s + (−5.36 + 5.36i)11-s + (16.8 − 7.30i)12-s + (11.0 − 11.0i)13-s + (5.19 + 7.63i)14-s + 0.381i·15-s + (−11.6 − 10.9i)16-s − 12.8·17-s + ⋯
L(s)  = 1  + (0.562 + 0.826i)2-s + (−1.08 − 1.08i)3-s + (−0.367 + 0.929i)4-s + (−0.0117 − 0.0117i)5-s + (0.286 − 1.50i)6-s + 0.659·7-s + (−0.975 + 0.218i)8-s + 1.34i·9-s + (0.00310 − 0.0162i)10-s + (−0.487 + 0.487i)11-s + (1.40 − 0.608i)12-s + (0.850 − 0.850i)13-s + (0.370 + 0.545i)14-s + 0.0254i·15-s + (−0.729 − 0.683i)16-s − 0.757·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.412i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.911 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.911 - 0.412i$
Analytic conductor: \(0.435968\)
Root analytic conductor: \(0.660279\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :1),\ 0.911 - 0.412i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.770378 + 0.166251i\)
\(L(\frac12)\) \(\approx\) \(0.770378 + 0.166251i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.12 - 1.65i)T \)
good3 \( 1 + (3.24 + 3.24i)T + 9iT^{2} \)
5 \( 1 + (0.0586 + 0.0586i)T + 25iT^{2} \)
7 \( 1 - 4.61T + 49T^{2} \)
11 \( 1 + (5.36 - 5.36i)T - 121iT^{2} \)
13 \( 1 + (-11.0 + 11.0i)T - 169iT^{2} \)
17 \( 1 + 12.8T + 289T^{2} \)
19 \( 1 + (-2.63 - 2.63i)T + 361iT^{2} \)
23 \( 1 - 16.3T + 529T^{2} \)
29 \( 1 + (26.0 - 26.0i)T - 841iT^{2} \)
31 \( 1 - 20.2iT - 961T^{2} \)
37 \( 1 + (-41.2 - 41.2i)T + 1.36e3iT^{2} \)
41 \( 1 - 3.29iT - 1.68e3T^{2} \)
43 \( 1 + (0.786 - 0.786i)T - 1.84e3iT^{2} \)
47 \( 1 + 79.7iT - 2.20e3T^{2} \)
53 \( 1 + (-1.06 - 1.06i)T + 2.80e3iT^{2} \)
59 \( 1 + (-32.5 + 32.5i)T - 3.48e3iT^{2} \)
61 \( 1 + (-15.2 + 15.2i)T - 3.72e3iT^{2} \)
67 \( 1 + (60.0 + 60.0i)T + 4.48e3iT^{2} \)
71 \( 1 - 56.3T + 5.04e3T^{2} \)
73 \( 1 + 9.70iT - 5.32e3T^{2} \)
79 \( 1 - 84.4iT - 6.24e3T^{2} \)
83 \( 1 + (-26.7 - 26.7i)T + 6.88e3iT^{2} \)
89 \( 1 - 115. iT - 7.92e3T^{2} \)
97 \( 1 + 146.T + 9.40e3T^{2} \)
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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

−18.25026686546723121479526920644, −17.85935043490362589130870728030, −16.58757012694145204434766652541, −15.14297884682136937507242627939, −13.44435969730640401330820760742, −12.50826909911028887636754102960, −11.15029139372647895587989645814, −8.129835113095462546847915139108, −6.71654148449982213536092582970, −5.23075575454201167238505510963, 4.30895712567018791429387203306, 5.77183978837161210226899607114, 9.327656388993218458874962249077, 11.01739937503836028067112955293, 11.36070723735327047177036977865, 13.31060993817031147739660821667, 14.95277598025408430863205856775, 16.15022386549459579473752660180, 17.61507081291335567722783411661, 18.93573590307263258487804229532

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