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Properties

Label 2-2e4-16.5-c7-0-1
Degree $2$
Conductor $16$
Sign $-0.235 - 0.971i$
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  + (−5.91 − 9.64i)2-s + (2.27 − 2.27i)3-s + (−57.9 + 114. i)4-s + (−242. − 242. i)5-s + (−35.3 − 8.47i)6-s + 1.64e3i·7-s + (1.44e3 − 115. i)8-s + 2.17e3i·9-s + (−903. + 3.77e3i)10-s + (−3.82e3 − 3.82e3i)11-s + (127. + 391. i)12-s + (−2.48e3 + 2.48e3i)13-s + (1.58e4 − 9.71e3i)14-s − 1.10e3·15-s + (−9.65e3 − 1.32e4i)16-s − 2.87e4·17-s + ⋯
L(s)  = 1  + (−0.522 − 0.852i)2-s + (0.0486 − 0.0486i)3-s + (−0.453 + 0.891i)4-s + (−0.867 − 0.867i)5-s + (−0.0668 − 0.0160i)6-s + 1.81i·7-s + (0.996 − 0.0800i)8-s + 0.995i·9-s + (−0.285 + 1.19i)10-s + (−0.866 − 0.866i)11-s + (0.0213 + 0.0653i)12-s + (−0.314 + 0.314i)13-s + (1.54 − 0.946i)14-s − 0.0843·15-s + (−0.589 − 0.807i)16-s − 1.41·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.235 - 0.971i$
Analytic conductor: \(4.99816\)
Root analytic conductor: \(2.23565\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :7/2),\ -0.235 - 0.971i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.175960 + 0.223702i\)
\(L(\frac12)\) \(\approx\) \(0.175960 + 0.223702i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.91 + 9.64i)T \)
good3 \( 1 + (-2.27 + 2.27i)T - 2.18e3iT^{2} \)
5 \( 1 + (242. + 242. i)T + 7.81e4iT^{2} \)
7 \( 1 - 1.64e3iT - 8.23e5T^{2} \)
11 \( 1 + (3.82e3 + 3.82e3i)T + 1.94e7iT^{2} \)
13 \( 1 + (2.48e3 - 2.48e3i)T - 6.27e7iT^{2} \)
17 \( 1 + 2.87e4T + 4.10e8T^{2} \)
19 \( 1 + (-3.39e3 + 3.39e3i)T - 8.93e8iT^{2} \)
23 \( 1 + 3.28e4iT - 3.40e9T^{2} \)
29 \( 1 + (1.14e5 - 1.14e5i)T - 1.72e10iT^{2} \)
31 \( 1 - 1.10e5T + 2.75e10T^{2} \)
37 \( 1 + (-1.48e5 - 1.48e5i)T + 9.49e10iT^{2} \)
41 \( 1 - 1.27e5iT - 1.94e11T^{2} \)
43 \( 1 + (2.62e5 + 2.62e5i)T + 2.71e11iT^{2} \)
47 \( 1 - 3.40e5T + 5.06e11T^{2} \)
53 \( 1 + (-7.55e5 - 7.55e5i)T + 1.17e12iT^{2} \)
59 \( 1 + (7.99e5 + 7.99e5i)T + 2.48e12iT^{2} \)
61 \( 1 + (3.79e5 - 3.79e5i)T - 3.14e12iT^{2} \)
67 \( 1 + (-1.12e6 + 1.12e6i)T - 6.06e12iT^{2} \)
71 \( 1 - 4.20e6iT - 9.09e12T^{2} \)
73 \( 1 + 5.28e6iT - 1.10e13T^{2} \)
79 \( 1 - 3.70e6T + 1.92e13T^{2} \)
83 \( 1 + (8.66e5 - 8.66e5i)T - 2.71e13iT^{2} \)
89 \( 1 - 6.21e6iT - 4.42e13T^{2} \)
97 \( 1 + 1.43e6T + 8.07e13T^{2} \)
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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

−18.34274082564545279765775039632, −16.53292367933459645856766121447, −15.60488839017218002205361309355, −13.26715006755902090185597774288, −12.16510208569080363572566349119, −11.05686286344722208223356094513, −8.935687822037638272356975068909, −8.169475974842448877448988534420, −4.92334768022351388423616622584, −2.44429758281354282910412481330, 0.20233034266791072277555041397, 4.15864911794830799542285182117, 6.87448357948447824787714782967, 7.68144331057595256854790318529, 9.879190176346397983216983604458, 11.02577538759251865292369543953, 13.38257167159286869076487910560, 14.84700693568468201841332038948, 15.63606334117035042109305822832, 17.24962605004983923621396832184

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