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Properties

Label 2-21-7.4-c5-0-2
Degree $2$
Conductor $21$
Sign $0.598 - 0.800i$
Analytic cond. $3.36806$
Root an. cond. $1.83522$
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  + (0.395 + 0.684i)2-s + (−4.5 + 7.79i)3-s + (15.6 − 27.1i)4-s + (52.0 + 90.2i)5-s − 7.11·6-s + (−7.12 + 129. i)7-s + 50.1·8-s + (−40.5 − 70.1i)9-s + (−41.1 + 71.3i)10-s + (248. − 430. i)11-s + (141. + 244. i)12-s − 206.·13-s + (−91.4 + 46.3i)14-s − 937.·15-s + (−482. − 835. i)16-s + (−31.5 + 54.6i)17-s + ⋯
L(s)  = 1  + (0.0698 + 0.121i)2-s + (−0.288 + 0.499i)3-s + (0.490 − 0.849i)4-s + (0.931 + 1.61i)5-s − 0.0807·6-s + (−0.0549 + 0.998i)7-s + 0.276·8-s + (−0.166 − 0.288i)9-s + (−0.130 + 0.225i)10-s + (0.620 − 1.07i)11-s + (0.283 + 0.490i)12-s − 0.338·13-s + (−0.124 + 0.0631i)14-s − 1.07·15-s + (−0.470 − 0.815i)16-s + (−0.0265 + 0.0459i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 - 0.800i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.598 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.598 - 0.800i$
Analytic conductor: \(3.36806\)
Root analytic conductor: \(1.83522\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :5/2),\ 0.598 - 0.800i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.45349 + 0.728115i\)
\(L(\frac12)\) \(\approx\) \(1.45349 + 0.728115i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.5 - 7.79i)T \)
7 \( 1 + (7.12 - 129. i)T \)
good2 \( 1 + (-0.395 - 0.684i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (-52.0 - 90.2i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-248. + 430. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 206.T + 3.71e5T^{2} \)
17 \( 1 + (31.5 - 54.6i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (661. + 1.14e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-97.2 - 168. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 4.32e3T + 2.05e7T^{2} \)
31 \( 1 + (-3.76e3 + 6.51e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (5.17e3 + 8.96e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 4.18e3T + 1.15e8T^{2} \)
43 \( 1 - 5.96e3T + 1.47e8T^{2} \)
47 \( 1 + (-2.19e3 - 3.80e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (8.89e3 - 1.54e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.75e3 - 3.03e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-5.31e3 - 9.20e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-6.63e3 + 1.14e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 3.88e4T + 1.80e9T^{2} \)
73 \( 1 + (1.56e4 - 2.71e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (1.97e4 + 3.42e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 1.02e5T + 3.93e9T^{2} \)
89 \( 1 + (-5.64e4 - 9.77e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 3.03e4T + 8.58e9T^{2} \)
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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

−17.37164416963853380741332962745, −15.71101599820335372435914677310, −14.79958755786609559070040512264, −13.91823846751875667812912224513, −11.50522390282443738026245206204, −10.59271850447395691654774767160, −9.390946790717921086278120514607, −6.58339595587481121657957889400, −5.73984646498827808965324466086, −2.59835994850349768932952320178, 1.55016034279135827785323409561, 4.60107374630584079775977856462, 6.74610556892610222850328816041, 8.351786685131296191529393069833, 10.05643030932958815490523016881, 12.09063940833894277972893176428, 12.76403088377064667794202476394, 13.88868287541100227381692211855, 16.17612161115586205137162099838, 17.21027849372280246176883621200

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