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Properties

Label 2-2e4-16.13-c5-0-2
Degree $2$
Conductor $16$
Sign $0.979 + 0.201i$
Analytic cond. $2.56614$
Root an. cond. $1.60191$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.36 − 4.54i)2-s + (16.8 + 16.8i)3-s + (−9.38 + 30.5i)4-s + (66.0 − 66.0i)5-s + (20.0 − 133. i)6-s + 75.3i·7-s + (170. − 60.2i)8-s + 327. i·9-s + (−522. − 78.2i)10-s + (−79.0 + 79.0i)11-s + (−675. + 358. i)12-s + (−238. − 238. i)13-s + (342. − 253. i)14-s + 2.23e3·15-s + (−847. − 574. i)16-s − 1.75e3·17-s + ⋯
L(s)  = 1  + (−0.594 − 0.804i)2-s + (1.08 + 1.08i)3-s + (−0.293 + 0.956i)4-s + (1.18 − 1.18i)5-s + (0.227 − 1.51i)6-s + 0.580i·7-s + (0.943 − 0.332i)8-s + 1.34i·9-s + (−1.65 − 0.247i)10-s + (−0.197 + 0.197i)11-s + (−1.35 + 0.718i)12-s + (−0.391 − 0.391i)13-s + (0.467 − 0.345i)14-s + 2.55·15-s + (−0.828 − 0.560i)16-s − 1.47·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.979 + 0.201i$
Analytic conductor: \(2.56614\)
Root analytic conductor: \(1.60191\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5/2),\ 0.979 + 0.201i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.45190 - 0.147426i\)
\(L(\frac12)\) \(\approx\) \(1.45190 - 0.147426i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.36 + 4.54i)T \)
good3 \( 1 + (-16.8 - 16.8i)T + 243iT^{2} \)
5 \( 1 + (-66.0 + 66.0i)T - 3.12e3iT^{2} \)
7 \( 1 - 75.3iT - 1.68e4T^{2} \)
11 \( 1 + (79.0 - 79.0i)T - 1.61e5iT^{2} \)
13 \( 1 + (238. + 238. i)T + 3.71e5iT^{2} \)
17 \( 1 + 1.75e3T + 1.41e6T^{2} \)
19 \( 1 + (311. + 311. i)T + 2.47e6iT^{2} \)
23 \( 1 + 1.30e3iT - 6.43e6T^{2} \)
29 \( 1 + (-2.58e3 - 2.58e3i)T + 2.05e7iT^{2} \)
31 \( 1 + 2.37e3T + 2.86e7T^{2} \)
37 \( 1 + (1.94e3 - 1.94e3i)T - 6.93e7iT^{2} \)
41 \( 1 + 3.91e3iT - 1.15e8T^{2} \)
43 \( 1 + (8.82e3 - 8.82e3i)T - 1.47e8iT^{2} \)
47 \( 1 - 2.32e4T + 2.29e8T^{2} \)
53 \( 1 + (-7.57e3 + 7.57e3i)T - 4.18e8iT^{2} \)
59 \( 1 + (3.30e4 - 3.30e4i)T - 7.14e8iT^{2} \)
61 \( 1 + (2.65e4 + 2.65e4i)T + 8.44e8iT^{2} \)
67 \( 1 + (-1.62e4 - 1.62e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 3.36e4iT - 1.80e9T^{2} \)
73 \( 1 - 5.62e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.29e3T + 3.07e9T^{2} \)
83 \( 1 + (-7.87e4 - 7.87e4i)T + 3.93e9iT^{2} \)
89 \( 1 + 8.69e4iT - 5.58e9T^{2} \)
97 \( 1 + 8.39e3T + 8.58e9T^{2} \)
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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

−18.04183857077556338942310878652, −16.83934378257137916947476103435, −15.52310235200364162020144960951, −13.76024669780051736972001380645, −12.60974939448344019752422483101, −10.39214730947159230854465824278, −9.231705832244961714987618988788, −8.630642809896823925426925905941, −4.69951836598634660717162508635, −2.37083630931323561727197754596, 2.11562578316265800102810159623, 6.44121087447635631979554002876, 7.41805753849027211773815548850, 9.099721454339143765453168463615, 10.56937935045767034382067538782, 13.51969117677511667442166438823, 13.96680686787847730723653240590, 15.13610460839243207197984041274, 17.24953405083272979785323212165, 18.17372333654813981943403970284

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