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Properties

Label 2-2e4-16.13-c13-0-24
Degree $2$
Conductor $16$
Sign $-0.390 - 0.920i$
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  + (34.4 − 83.6i)2-s + (−787. − 787. i)3-s + (−5.81e3 − 5.76e3i)4-s + (1.33e4 − 1.33e4i)5-s + (−9.30e4 + 3.87e4i)6-s − 5.47e5i·7-s + (−6.83e5 + 2.87e5i)8-s − 3.54e5i·9-s + (−6.58e5 − 1.58e6i)10-s + (−4.63e6 + 4.63e6i)11-s + (3.74e4 + 9.12e6i)12-s + (4.58e6 + 4.58e6i)13-s + (−4.57e7 − 1.88e7i)14-s − 2.10e7·15-s + (5.51e5 + 6.71e7i)16-s + 1.39e8·17-s + ⋯
L(s)  = 1  + (0.380 − 0.924i)2-s + (−0.623 − 0.623i)3-s + (−0.710 − 0.704i)4-s + (0.382 − 0.382i)5-s + (−0.814 + 0.339i)6-s − 1.75i·7-s + (−0.921 + 0.388i)8-s − 0.222i·9-s + (−0.208 − 0.499i)10-s + (−0.788 + 0.788i)11-s + (0.00362 + 0.881i)12-s + (0.263 + 0.263i)13-s + (−1.62 − 0.669i)14-s − 0.477·15-s + (0.00821 + 0.999i)16-s + 1.40·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.390 - 0.920i$
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: \((2,\ 16,\ (\ :13/2),\ -0.390 - 0.920i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.639065 + 0.964987i\)
\(L(\frac12)\) \(\approx\) \(0.639065 + 0.964987i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-34.4 + 83.6i)T \)
good3 \( 1 + (787. + 787. i)T + 1.59e6iT^{2} \)
5 \( 1 + (-1.33e4 + 1.33e4i)T - 1.22e9iT^{2} \)
7 \( 1 + 5.47e5iT - 9.68e10T^{2} \)
11 \( 1 + (4.63e6 - 4.63e6i)T - 3.45e13iT^{2} \)
13 \( 1 + (-4.58e6 - 4.58e6i)T + 3.02e14iT^{2} \)
17 \( 1 - 1.39e8T + 9.90e15T^{2} \)
19 \( 1 + (-9.33e7 - 9.33e7i)T + 4.20e16iT^{2} \)
23 \( 1 + 6.57e8iT - 5.04e17T^{2} \)
29 \( 1 + (1.55e9 + 1.55e9i)T + 1.02e19iT^{2} \)
31 \( 1 + 6.80e9T + 2.44e19T^{2} \)
37 \( 1 + (-5.41e9 + 5.41e9i)T - 2.43e20iT^{2} \)
41 \( 1 + 2.37e10iT - 9.25e20T^{2} \)
43 \( 1 + (-5.22e10 + 5.22e10i)T - 1.71e21iT^{2} \)
47 \( 1 + 6.79e10T + 5.46e21T^{2} \)
53 \( 1 + (8.05e9 - 8.05e9i)T - 2.60e22iT^{2} \)
59 \( 1 + (-2.50e10 + 2.50e10i)T - 1.04e23iT^{2} \)
61 \( 1 + (-2.20e11 - 2.20e11i)T + 1.61e23iT^{2} \)
67 \( 1 + (-3.27e11 - 3.27e11i)T + 5.48e23iT^{2} \)
71 \( 1 + 1.50e12iT - 1.16e24T^{2} \)
73 \( 1 - 4.75e11iT - 1.67e24T^{2} \)
79 \( 1 + 1.24e12T + 4.66e24T^{2} \)
83 \( 1 + (3.02e12 + 3.02e12i)T + 8.87e24iT^{2} \)
89 \( 1 - 7.21e12iT - 2.19e25T^{2} \)
97 \( 1 + 3.17e12T + 6.73e25T^{2} \)
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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

−14.45592356791328687512670893212, −13.24379932157878175125159322242, −12.37234048485251396249512184403, −10.84929144934509171401246126236, −9.735624147287114483966936122683, −7.33881609647125322584770311294, −5.50879940750824804753463023304, −3.84096334171308492914416593363, −1.50624012719809244620455221081, −0.44365638109537998762292906830, 2.98253882317177464891629667292, 5.32186570037028317243025800317, 5.82422745417265896335634419728, 8.045233854094092642793460073877, 9.588651773005212698904013352061, 11.39638858788280735653277067967, 12.89685385317407593548862573886, 14.45257638249785600436551861231, 15.69025507779300697471935818632, 16.38077246076100920912983610326

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