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Properties

Label 2-21-3.2-c4-0-6
Degree $2$
Conductor $21$
Sign $-0.588 + 0.808i$
Analytic cond. $2.17076$
Root an. cond. $1.47335$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.97i·2-s + (5.29 − 7.27i)3-s − 19.6·4-s + 22.9i·5-s + (−43.4 − 31.6i)6-s + 18.5·7-s + 21.7i·8-s + (−24.9 − 77.0i)9-s + 137.·10-s + 105. i·11-s + (−103. + 143. i)12-s + 281.·13-s − 110. i·14-s + (167. + 121. i)15-s − 184.·16-s − 81.9i·17-s + ⋯
L(s)  = 1  − 1.49i·2-s + (0.588 − 0.808i)3-s − 1.22·4-s + 0.918i·5-s + (−1.20 − 0.877i)6-s + 0.377·7-s + 0.340i·8-s + (−0.308 − 0.951i)9-s + 1.37·10-s + 0.873i·11-s + (−0.722 + 0.993i)12-s + 1.66·13-s − 0.564i·14-s + (0.743 + 0.540i)15-s − 0.719·16-s − 0.283i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.588 + 0.808i$
Analytic conductor: \(2.17076\)
Root analytic conductor: \(1.47335\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :2),\ -0.588 + 0.808i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.667937 - 1.31147i\)
\(L(\frac12)\) \(\approx\) \(0.667937 - 1.31147i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.29 + 7.27i)T \)
7 \( 1 - 18.5T \)
good2 \( 1 + 5.97iT - 16T^{2} \)
5 \( 1 - 22.9iT - 625T^{2} \)
11 \( 1 - 105. iT - 1.46e4T^{2} \)
13 \( 1 - 281.T + 2.85e4T^{2} \)
17 \( 1 + 81.9iT - 8.35e4T^{2} \)
19 \( 1 + 467.T + 1.30e5T^{2} \)
23 \( 1 - 625. iT - 2.79e5T^{2} \)
29 \( 1 - 806. iT - 7.07e5T^{2} \)
31 \( 1 - 198.T + 9.23e5T^{2} \)
37 \( 1 + 1.82e3T + 1.87e6T^{2} \)
41 \( 1 + 580. iT - 2.82e6T^{2} \)
43 \( 1 + 287.T + 3.41e6T^{2} \)
47 \( 1 + 3.07e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.03e3iT - 7.89e6T^{2} \)
59 \( 1 + 439. iT - 1.21e7T^{2} \)
61 \( 1 - 1.83e3T + 1.38e7T^{2} \)
67 \( 1 + 4.27e3T + 2.01e7T^{2} \)
71 \( 1 + 5.14e3iT - 2.54e7T^{2} \)
73 \( 1 - 3.47e3T + 2.83e7T^{2} \)
79 \( 1 + 8.63e3T + 3.89e7T^{2} \)
83 \( 1 - 2.12e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.46e3iT - 6.27e7T^{2} \)
97 \( 1 - 4.82e3T + 8.85e7T^{2} \)
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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

−17.81679771538370686730978291064, −15.27027137926841517555016210264, −13.92411063009411013903300819286, −12.86640380457333542937033099404, −11.57350247661094016605048936765, −10.46338056418504009546168365917, −8.780014365102236193723719957322, −6.86614255847059640845768959461, −3.50295475604614218864400523024, −1.80301479030275643572363310825, 4.41900249527205913815427093213, 6.00177983993118042301691130747, 8.377534191316208802675623087843, 8.697963629401132713014782978097, 10.89420846632357141345979644164, 13.28147603063516659862589791846, 14.34049898785812642167232342294, 15.55635276268960625145062796170, 16.33244365971267346539341375305, 17.18526054830599652802048705144

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