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Properties

Label 2-2e4-16.13-c13-0-7
Degree $2$
Conductor $16$
Sign $0.627 - 0.778i$
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  + (−71.4 − 55.5i)2-s + (293. + 293. i)3-s + (2.02e3 + 7.93e3i)4-s + (2.27e4 − 2.27e4i)5-s + (−4.67e3 − 3.72e4i)6-s + 3.43e5i·7-s + (2.96e5 − 6.79e5i)8-s − 1.42e6i·9-s + (−2.88e6 + 3.61e5i)10-s + (−7.55e6 + 7.55e6i)11-s + (−1.73e6 + 2.92e6i)12-s + (−6.02e6 − 6.02e6i)13-s + (1.90e7 − 2.45e7i)14-s + 1.33e7·15-s + (−5.89e7 + 3.21e7i)16-s + 1.22e8·17-s + ⋯
L(s)  = 1  + (−0.789 − 0.613i)2-s + (0.232 + 0.232i)3-s + (0.247 + 0.969i)4-s + (0.650 − 0.650i)5-s + (−0.0409 − 0.326i)6-s + 1.10i·7-s + (0.399 − 0.916i)8-s − 0.891i·9-s + (−0.912 + 0.114i)10-s + (−1.28 + 1.28i)11-s + (−0.167 + 0.282i)12-s + (−0.346 − 0.346i)13-s + (0.677 − 0.871i)14-s + 0.302·15-s + (−0.877 + 0.478i)16-s + 1.23·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.627 - 0.778i$
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: \((2,\ 16,\ (\ :13/2),\ 0.627 - 0.778i)\)

Particular Values

\(L(7)\) \(\approx\) \(1.09279 + 0.522449i\)
\(L(\frac12)\) \(\approx\) \(1.09279 + 0.522449i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (71.4 + 55.5i)T \)
good3 \( 1 + (-293. - 293. i)T + 1.59e6iT^{2} \)
5 \( 1 + (-2.27e4 + 2.27e4i)T - 1.22e9iT^{2} \)
7 \( 1 - 3.43e5iT - 9.68e10T^{2} \)
11 \( 1 + (7.55e6 - 7.55e6i)T - 3.45e13iT^{2} \)
13 \( 1 + (6.02e6 + 6.02e6i)T + 3.02e14iT^{2} \)
17 \( 1 - 1.22e8T + 9.90e15T^{2} \)
19 \( 1 + (-1.66e8 - 1.66e8i)T + 4.20e16iT^{2} \)
23 \( 1 - 7.77e8iT - 5.04e17T^{2} \)
29 \( 1 + (-3.04e9 - 3.04e9i)T + 1.02e19iT^{2} \)
31 \( 1 + 6.68e9T + 2.44e19T^{2} \)
37 \( 1 + (-7.06e9 + 7.06e9i)T - 2.43e20iT^{2} \)
41 \( 1 - 4.77e10iT - 9.25e20T^{2} \)
43 \( 1 + (3.20e10 - 3.20e10i)T - 1.71e21iT^{2} \)
47 \( 1 - 4.56e10T + 5.46e21T^{2} \)
53 \( 1 + (5.36e10 - 5.36e10i)T - 2.60e22iT^{2} \)
59 \( 1 + (-3.79e10 + 3.79e10i)T - 1.04e23iT^{2} \)
61 \( 1 + (-2.48e11 - 2.48e11i)T + 1.61e23iT^{2} \)
67 \( 1 + (-6.55e10 - 6.55e10i)T + 5.48e23iT^{2} \)
71 \( 1 + 4.07e11iT - 1.16e24T^{2} \)
73 \( 1 + 1.83e12iT - 1.67e24T^{2} \)
79 \( 1 - 8.45e11T + 4.66e24T^{2} \)
83 \( 1 + (2.16e11 + 2.16e11i)T + 8.87e24iT^{2} \)
89 \( 1 + 2.96e12iT - 2.19e25T^{2} \)
97 \( 1 + 6.69e12T + 6.73e25T^{2} \)
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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

−16.25268410795299355940491958031, −14.99274020404468778726325594822, −12.82578966675561520889671512313, −12.09698039987917548461516415600, −9.985471624379075015717510364646, −9.292924835226435111192603518757, −7.74417466170343479375330067914, −5.35807712202168388173198316404, −3.01563702516700592652727643577, −1.48422717509789645606293261092, 0.60502684225555246080773052894, 2.52296937585220626968759242599, 5.41070325307473807787009516538, 7.07306180501129163916770315836, 8.171623358357162720303783488608, 10.12087732118623922494017718862, 10.86065741403413562968146799966, 13.63949095368757353192100364041, 14.18700396777999132007084807537, 16.06079520622396748145367697467

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