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Properties

Label 2-2e4-16.5-c13-0-8
Degree $2$
Conductor $16$
Sign $-0.721 - 0.691i$
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  + (90.1 + 8.39i)2-s + (−1.27e3 + 1.27e3i)3-s + (8.05e3 + 1.51e3i)4-s + (3.64e4 + 3.64e4i)5-s + (−1.25e5 + 1.04e5i)6-s − 6.80e4i·7-s + (7.12e5 + 2.03e5i)8-s − 1.64e6i·9-s + (2.98e6 + 3.59e6i)10-s + (−1.69e6 − 1.69e6i)11-s + (−1.21e7 + 8.32e6i)12-s + (−2.34e7 + 2.34e7i)13-s + (5.71e5 − 6.13e6i)14-s − 9.28e7·15-s + (6.25e7 + 2.43e7i)16-s + 6.73e7·17-s + ⋯
L(s)  = 1  + (0.995 + 0.0927i)2-s + (−1.00 + 1.00i)3-s + (0.982 + 0.184i)4-s + (1.04 + 1.04i)5-s + (−1.09 + 0.910i)6-s − 0.218i·7-s + (0.961 + 0.275i)8-s − 1.03i·9-s + (0.943 + 1.13i)10-s + (−0.288 − 0.288i)11-s + (−1.17 + 0.804i)12-s + (−1.34 + 1.34i)13-s + (0.0202 − 0.217i)14-s − 2.10·15-s + (0.931 + 0.363i)16-s + 0.676·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.721 - 0.691i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.721 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.721 - 0.691i$
Analytic conductor: \(17.1569\)
Root analytic conductor: \(4.14209\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :13/2),\ -0.721 - 0.691i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.987058 + 2.45610i\)
\(L(\frac12)\) \(\approx\) \(0.987058 + 2.45610i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-90.1 - 8.39i)T \)
good3 \( 1 + (1.27e3 - 1.27e3i)T - 1.59e6iT^{2} \)
5 \( 1 + (-3.64e4 - 3.64e4i)T + 1.22e9iT^{2} \)
7 \( 1 + 6.80e4iT - 9.68e10T^{2} \)
11 \( 1 + (1.69e6 + 1.69e6i)T + 3.45e13iT^{2} \)
13 \( 1 + (2.34e7 - 2.34e7i)T - 3.02e14iT^{2} \)
17 \( 1 - 6.73e7T + 9.90e15T^{2} \)
19 \( 1 + (1.39e8 - 1.39e8i)T - 4.20e16iT^{2} \)
23 \( 1 + 1.17e9iT - 5.04e17T^{2} \)
29 \( 1 + (2.78e9 - 2.78e9i)T - 1.02e19iT^{2} \)
31 \( 1 - 6.84e9T + 2.44e19T^{2} \)
37 \( 1 + (2.23e9 + 2.23e9i)T + 2.43e20iT^{2} \)
41 \( 1 + 1.99e9iT - 9.25e20T^{2} \)
43 \( 1 + (-1.50e10 - 1.50e10i)T + 1.71e21iT^{2} \)
47 \( 1 - 5.49e10T + 5.46e21T^{2} \)
53 \( 1 + (-1.22e11 - 1.22e11i)T + 2.60e22iT^{2} \)
59 \( 1 + (-1.58e11 - 1.58e11i)T + 1.04e23iT^{2} \)
61 \( 1 + (-2.28e11 + 2.28e11i)T - 1.61e23iT^{2} \)
67 \( 1 + (1.02e11 - 1.02e11i)T - 5.48e23iT^{2} \)
71 \( 1 + 8.82e11iT - 1.16e24T^{2} \)
73 \( 1 - 1.62e12iT - 1.67e24T^{2} \)
79 \( 1 + 7.90e11T + 4.66e24T^{2} \)
83 \( 1 + (2.41e11 - 2.41e11i)T - 8.87e24iT^{2} \)
89 \( 1 - 3.92e12iT - 2.19e25T^{2} \)
97 \( 1 - 4.42e12T + 6.73e25T^{2} \)
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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

−16.48125572109893636620193595189, −14.83936520093946229432648329925, −14.05203977645311137771906321178, −12.18316785664145828875763826916, −10.79338124364575622062156257586, −10.04309580707693440963287642677, −6.82820995510964376154850309067, −5.69399789529013932951230466153, −4.34131831682022096206078980415, −2.42097809557448406751991359401, 0.810988045165575710982013643878, 2.21832634962205312504958971032, 5.14735130317867316220343751630, 5.80316068341727040331281140698, 7.46688364153477219657033831703, 10.02939920015208821811015264800, 11.84472686923192390474967236337, 12.75208935438955757994783871999, 13.43165675911198456565521452617, 15.28970426889157861315500563313

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