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Properties

Label 2-21-7.4-c5-0-0
Degree $2$
Conductor $21$
Sign $0.341 - 0.940i$
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  + (−5.11 − 8.85i)2-s + (−4.5 + 7.79i)3-s + (−36.2 + 62.8i)4-s + (−11.8 − 20.5i)5-s + 92.0·6-s + (30.6 + 125. i)7-s + 414.·8-s + (−40.5 − 70.1i)9-s + (−121. + 210. i)10-s + (−232. + 403. i)11-s + (−326. − 565. i)12-s − 1.01e3·13-s + (958. − 915. i)14-s + 213.·15-s + (−957. − 1.65e3i)16-s + (−280. + 486. i)17-s + ⋯
L(s)  = 1  + (−0.903 − 1.56i)2-s + (−0.288 + 0.499i)3-s + (−1.13 + 1.96i)4-s + (−0.212 − 0.367i)5-s + 1.04·6-s + (0.236 + 0.971i)7-s + 2.28·8-s + (−0.166 − 0.288i)9-s + (−0.383 + 0.665i)10-s + (−0.580 + 1.00i)11-s + (−0.654 − 1.13i)12-s − 1.67·13-s + (1.30 − 1.24i)14-s + 0.245·15-s + (−0.935 − 1.61i)16-s + (−0.235 + 0.408i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 - 0.940i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.341 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.341 - 0.940i$
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: \((2,\ 21,\ (\ :5/2),\ 0.341 - 0.940i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.235494 + 0.165078i\)
\(L(\frac12)\) \(\approx\) \(0.235494 + 0.165078i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-30.6 - 125. i)T \)
good2 \( 1 + (5.11 + 8.85i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (11.8 + 20.5i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (232. - 403. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 1.01e3T + 3.71e5T^{2} \)
17 \( 1 + (280. - 486. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-693. - 1.20e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (2.05e3 + 3.56e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 2.38e3T + 2.05e7T^{2} \)
31 \( 1 + (1.47e3 - 2.55e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-4.95e3 - 8.58e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 4.47e3T + 1.15e8T^{2} \)
43 \( 1 - 5.18e3T + 1.47e8T^{2} \)
47 \( 1 + (-1.56e3 - 2.70e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (570. - 988. i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.37e4 - 2.38e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.05e4 - 1.82e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-2.77e4 + 4.81e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 6.07e3T + 1.80e9T^{2} \)
73 \( 1 + (-8.38e3 + 1.45e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-2.42e3 - 4.19e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 6.01e4T + 3.93e9T^{2} \)
89 \( 1 + (-3.12e4 - 5.41e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 6.36e4T + 8.58e9T^{2} \)
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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

−17.74530542142103435184459712091, −16.55118276064083122127299158183, −14.88383781527088682716086942689, −12.44514274911028901307442877560, −12.09938114377609983444203564754, −10.45067917021049627520293797349, −9.506765308451379386004801898952, −8.113125646187576267229409539109, −4.67259239994761342123715478166, −2.36299147365530659116393497233, 0.28155428817559639907044665997, 5.35191017856911452163792293594, 7.11009184049610254783570598158, 7.76346036288667338088240328213, 9.610485331536739691646612393951, 11.13318506035209393954303444479, 13.50314498711475112594532765746, 14.56434737488541730233843468879, 15.89678226819538714757303367300, 16.99024184990649280752339102823

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