Properties

Label 2-2e4-16.13-c13-0-11
Degree $2$
Conductor $16$
Sign $0.640 - 0.768i$
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  + (−55.8 + 71.2i)2-s + (375. + 375. i)3-s + (−1.96e3 − 7.95e3i)4-s + (−1.14e4 + 1.14e4i)5-s + (−4.77e4 + 5.79e3i)6-s − 9.99e3i·7-s + (6.76e5 + 3.04e5i)8-s − 1.31e6i·9-s + (−1.77e5 − 1.45e6i)10-s + (3.95e6 − 3.95e6i)11-s + (2.25e6 − 3.72e6i)12-s + (−1.08e6 − 1.08e6i)13-s + (7.11e5 + 5.57e5i)14-s − 8.62e6·15-s + (−5.94e7 + 3.11e7i)16-s + 6.58e7·17-s + ⋯
L(s)  = 1  + (−0.616 + 0.787i)2-s + (0.297 + 0.297i)3-s + (−0.239 − 0.970i)4-s + (−0.328 + 0.328i)5-s + (−0.417 + 0.0507i)6-s − 0.0320i·7-s + (0.911 + 0.410i)8-s − 0.822i·9-s + (−0.0559 − 0.461i)10-s + (0.672 − 0.672i)11-s + (0.217 − 0.360i)12-s + (−0.0625 − 0.0625i)13-s + (0.0252 + 0.0197i)14-s − 0.195·15-s + (−0.885 + 0.464i)16-s + 0.661·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.768i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.640 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.640 - 0.768i$
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: \((2,\ 16,\ (\ :13/2),\ 0.640 - 0.768i)\)

Particular Values

\(L(7)\) \(\approx\) \(1.30528 + 0.611342i\)
\(L(\frac12)\) \(\approx\) \(1.30528 + 0.611342i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (55.8 - 71.2i)T \)
good3 \( 1 + (-375. - 375. i)T + 1.59e6iT^{2} \)
5 \( 1 + (1.14e4 - 1.14e4i)T - 1.22e9iT^{2} \)
7 \( 1 + 9.99e3iT - 9.68e10T^{2} \)
11 \( 1 + (-3.95e6 + 3.95e6i)T - 3.45e13iT^{2} \)
13 \( 1 + (1.08e6 + 1.08e6i)T + 3.02e14iT^{2} \)
17 \( 1 - 6.58e7T + 9.90e15T^{2} \)
19 \( 1 + (-1.88e8 - 1.88e8i)T + 4.20e16iT^{2} \)
23 \( 1 - 5.33e8iT - 5.04e17T^{2} \)
29 \( 1 + (-3.41e8 - 3.41e8i)T + 1.02e19iT^{2} \)
31 \( 1 - 3.93e9T + 2.44e19T^{2} \)
37 \( 1 + (-1.64e9 + 1.64e9i)T - 2.43e20iT^{2} \)
41 \( 1 + 1.36e10iT - 9.25e20T^{2} \)
43 \( 1 + (-3.19e10 + 3.19e10i)T - 1.71e21iT^{2} \)
47 \( 1 - 5.90e9T + 5.46e21T^{2} \)
53 \( 1 + (-1.15e11 + 1.15e11i)T - 2.60e22iT^{2} \)
59 \( 1 + (-5.51e10 + 5.51e10i)T - 1.04e23iT^{2} \)
61 \( 1 + (-4.95e11 - 4.95e11i)T + 1.61e23iT^{2} \)
67 \( 1 + (3.89e11 + 3.89e11i)T + 5.48e23iT^{2} \)
71 \( 1 + 7.06e11iT - 1.16e24T^{2} \)
73 \( 1 + 6.83e11iT - 1.67e24T^{2} \)
79 \( 1 + 3.34e12T + 4.66e24T^{2} \)
83 \( 1 + (3.07e12 + 3.07e12i)T + 8.87e24iT^{2} \)
89 \( 1 - 6.44e12iT - 2.19e25T^{2} \)
97 \( 1 - 1.54e13T + 6.73e25T^{2} \)
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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

−16.04441384004075131112936713288, −14.93352727866093267832681667573, −13.95920429315358256898284324018, −11.72427435037928320610938680200, −10.04802420813905198679579348539, −8.843568446544751287054457382786, −7.35211119222280899914778728928, −5.80404272204804873178098296514, −3.63192059283101333554890408580, −0.964437249044070332193177691778, 0.997379218008976190382447445025, 2.58425721659430546577382987823, 4.49218958146929116580990481263, 7.31475610706818540374094059189, 8.566892820685884593332130068761, 10.00321952707294465027654597631, 11.57191268828640039787822952346, 12.70680497795254571926822555740, 14.06844810094342595814394729295, 16.04403503497798303552764348964

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