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Properties

Label 2-2e4-16.13-c7-0-5
Degree $2$
Conductor $16$
Sign $0.126 + 0.991i$
Analytic cond. $4.99816$
Root an. cond. $2.23565$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−10.1 − 5.01i)2-s + (8.25 + 8.25i)3-s + (77.7 + 101. i)4-s + (63.0 − 63.0i)5-s + (−42.3 − 125. i)6-s − 847. i·7-s + (−277. − 1.42e3i)8-s − 2.05e3i·9-s + (−955. + 323. i)10-s + (2.94e3 − 2.94e3i)11-s + (−198. + 1.48e3i)12-s + (910. + 910. i)13-s + (−4.25e3 + 8.59e3i)14-s + 1.04e3·15-s + (−4.30e3 + 1.58e4i)16-s + 1.68e4·17-s + ⋯
L(s)  = 1  + (−0.896 − 0.443i)2-s + (0.176 + 0.176i)3-s + (0.607 + 0.794i)4-s + (0.225 − 0.225i)5-s + (−0.0799 − 0.236i)6-s − 0.934i·7-s + (−0.191 − 0.981i)8-s − 0.937i·9-s + (−0.302 + 0.102i)10-s + (0.666 − 0.666i)11-s + (−0.0331 + 0.247i)12-s + (0.114 + 0.114i)13-s + (−0.414 + 0.837i)14-s + 0.0796·15-s + (−0.262 + 0.964i)16-s + 0.834·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.126 + 0.991i$
Analytic conductor: \(4.99816\)
Root analytic conductor: \(2.23565\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :7/2),\ 0.126 + 0.991i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.804624 - 0.708717i\)
\(L(\frac12)\) \(\approx\) \(0.804624 - 0.708717i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (10.1 + 5.01i)T \)
good3 \( 1 + (-8.25 - 8.25i)T + 2.18e3iT^{2} \)
5 \( 1 + (-63.0 + 63.0i)T - 7.81e4iT^{2} \)
7 \( 1 + 847. iT - 8.23e5T^{2} \)
11 \( 1 + (-2.94e3 + 2.94e3i)T - 1.94e7iT^{2} \)
13 \( 1 + (-910. - 910. i)T + 6.27e7iT^{2} \)
17 \( 1 - 1.68e4T + 4.10e8T^{2} \)
19 \( 1 + (1.10e4 + 1.10e4i)T + 8.93e8iT^{2} \)
23 \( 1 + 6.76e4iT - 3.40e9T^{2} \)
29 \( 1 + (-7.27e4 - 7.27e4i)T + 1.72e10iT^{2} \)
31 \( 1 + 2.43e5T + 2.75e10T^{2} \)
37 \( 1 + (1.11e5 - 1.11e5i)T - 9.49e10iT^{2} \)
41 \( 1 - 8.59e5iT - 1.94e11T^{2} \)
43 \( 1 + (-1.08e5 + 1.08e5i)T - 2.71e11iT^{2} \)
47 \( 1 + 3.87e5T + 5.06e11T^{2} \)
53 \( 1 + (-1.29e6 + 1.29e6i)T - 1.17e12iT^{2} \)
59 \( 1 + (-1.70e6 + 1.70e6i)T - 2.48e12iT^{2} \)
61 \( 1 + (1.95e6 + 1.95e6i)T + 3.14e12iT^{2} \)
67 \( 1 + (-3.23e6 - 3.23e6i)T + 6.06e12iT^{2} \)
71 \( 1 - 3.07e6iT - 9.09e12T^{2} \)
73 \( 1 - 2.72e6iT - 1.10e13T^{2} \)
79 \( 1 + 2.89e6T + 1.92e13T^{2} \)
83 \( 1 + (-2.74e6 - 2.74e6i)T + 2.71e13iT^{2} \)
89 \( 1 - 2.91e6iT - 4.42e13T^{2} \)
97 \( 1 - 8.03e6T + 8.07e13T^{2} \)
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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

−17.19902143275778776449386769923, −16.35683337638688442015840697901, −14.54673679656296112732637988177, −12.82352006380262637450429517218, −11.30366661571238781698052932524, −9.879425485033139466996918909673, −8.615723467382185798111833795494, −6.76697461607006599077354336057, −3.59963826862410710046741107350, −0.943474408132383081074578198487, 2.01483339141988993441643442171, 5.69242039416159554140727523505, 7.48135263518803584754621551438, 8.964133635084294076231536063377, 10.39855908493668988509854056821, 12.06412922231822754305225082076, 14.10466550100659282853131204924, 15.29767332958949621642025525274, 16.60067381697358894070157132755, 17.87649158366379185896230130709

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