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Properties

Label 2-21-7.3-c6-0-4
Degree $2$
Conductor $21$
Sign $0.938 - 0.346i$
Analytic cond. $4.83113$
Root an. cond. $2.19798$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.80 + 3.12i)2-s + (13.5 + 7.79i)3-s + (25.4 − 44.1i)4-s + (71.9 − 41.5i)5-s + 56.2i·6-s + (77.0 + 334. i)7-s + 414.·8-s + (121.5 + 210. i)9-s + (259. + 149. i)10-s + (221. − 383. i)11-s + (688. − 397. i)12-s + 696. i·13-s + (−904. + 843. i)14-s + 1.29e3·15-s + (−883. − 1.53e3i)16-s + (−5.44e3 − 3.14e3i)17-s + ⋯
L(s)  = 1  + (0.225 + 0.390i)2-s + (0.5 + 0.288i)3-s + (0.398 − 0.690i)4-s + (0.575 − 0.332i)5-s + 0.260i·6-s + (0.224 + 0.974i)7-s + 0.810·8-s + (0.166 + 0.288i)9-s + (0.259 + 0.149i)10-s + (0.166 − 0.287i)11-s + (0.398 − 0.230i)12-s + 0.317i·13-s + (−0.329 + 0.307i)14-s + 0.383·15-s + (−0.215 − 0.373i)16-s + (−1.10 − 0.640i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.346i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.938 - 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.938 - 0.346i$
Analytic conductor: \(4.83113\)
Root analytic conductor: \(2.19798\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3),\ 0.938 - 0.346i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.26303 + 0.404136i\)
\(L(\frac12)\) \(\approx\) \(2.26303 + 0.404136i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-13.5 - 7.79i)T \)
7 \( 1 + (-77.0 - 334. i)T \)
good2 \( 1 + (-1.80 - 3.12i)T + (-32 + 55.4i)T^{2} \)
5 \( 1 + (-71.9 + 41.5i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (-221. + 383. i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 - 696. iT - 4.82e6T^{2} \)
17 \( 1 + (5.44e3 + 3.14e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (2.32e3 - 1.34e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (7.88e3 + 1.36e4i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + 2.32e4T + 5.94e8T^{2} \)
31 \( 1 + (-4.11e4 - 2.37e4i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-5.07e3 - 8.79e3i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 3.81e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.51e5T + 6.32e9T^{2} \)
47 \( 1 + (4.35e4 - 2.51e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (-9.97e4 + 1.72e5i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (-3.35e5 - 1.93e5i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (1.02e4 - 5.89e3i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-1.92e5 + 3.32e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 1.56e5T + 1.28e11T^{2} \)
73 \( 1 + (-3.25e5 - 1.87e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (1.65e4 + 2.87e4i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 9.84e5iT - 3.26e11T^{2} \)
89 \( 1 + (2.27e5 - 1.31e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 5.75e5iT - 8.32e11T^{2} \)
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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

−16.56270965691785323284418239638, −15.48215399207945731886409472959, −14.50446386057366380717892750769, −13.35723133526094397364076134727, −11.49985651740546567700959473670, −9.894488479715852686764606807461, −8.589419392436654947978968579292, −6.44405189219881082086790383539, −4.96454371995627302731878644212, −2.08587472078149138640685531628, 2.05946532982861359310637616495, 3.93885903225597496845093492019, 6.76106962101120126158108468666, 8.095403499820281522282737613740, 10.09953342764295974119559452149, 11.44994331964439743807276233176, 13.05380030179075276747864559540, 13.80404556805187924935494271377, 15.35585928249352104300795482125, 17.02220845672826497813508722912

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