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Properties

Label 2-21-21.20-c29-0-23
Degree $2$
Conductor $21$
Sign $-0.889 - 0.456i$
Analytic cond. $111.883$
Root an. cond. $10.5775$
Motivic weight $29$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.97e3i·2-s + (6.63e6 + 4.95e6i)3-s + 5.32e8·4-s − 2.10e10·5-s + (9.81e9 − 1.31e10i)6-s + (1.76e12 − 2.99e11i)7-s − 2.11e12i·8-s + (1.94e13 + 6.58e13i)9-s + 4.15e13i·10-s + 1.97e15i·11-s + (3.53e15 + 2.64e15i)12-s + 2.35e16i·13-s + (−5.93e14 − 3.50e15i)14-s + (−1.39e17 − 1.04e17i)15-s + 2.81e17·16-s − 4.82e17·17-s + ⋯
L(s)  = 1  − 0.0854i·2-s + (0.801 + 0.598i)3-s + 0.992·4-s − 1.53·5-s + (0.0511 − 0.0684i)6-s + (0.985 − 0.167i)7-s − 0.170i·8-s + (0.283 + 0.958i)9-s + 0.131i·10-s + 1.56i·11-s + (0.795 + 0.594i)12-s + 1.65i·13-s + (−0.0142 − 0.0842i)14-s + (−1.23 − 0.921i)15-s + 0.978·16-s − 0.694·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.889 - 0.456i$
Analytic conductor: \(111.883\)
Root analytic conductor: \(10.5775\)
Motivic weight: \(29\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :29/2),\ -0.889 - 0.456i)\)

Particular Values

\(L(15)\) \(\approx\) \(2.303974142\)
\(L(\frac12)\) \(\approx\) \(2.303974142\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-6.63e6 - 4.95e6i)T \)
7 \( 1 + (-1.76e12 + 2.99e11i)T \)
good2 \( 1 + 1.97e3iT - 5.36e8T^{2} \)
5 \( 1 + 2.10e10T + 1.86e20T^{2} \)
11 \( 1 - 1.97e15iT - 1.58e30T^{2} \)
13 \( 1 - 2.35e16iT - 2.01e32T^{2} \)
17 \( 1 + 4.82e17T + 4.81e35T^{2} \)
19 \( 1 - 5.67e17iT - 1.21e37T^{2} \)
23 \( 1 + 7.46e19iT - 3.09e39T^{2} \)
29 \( 1 + 1.34e21iT - 2.56e42T^{2} \)
31 \( 1 + 7.84e20iT - 1.77e43T^{2} \)
37 \( 1 + 8.59e22T + 3.00e45T^{2} \)
41 \( 1 - 1.02e23T + 5.89e46T^{2} \)
43 \( 1 - 2.29e23T + 2.34e47T^{2} \)
47 \( 1 - 1.10e21T + 3.09e48T^{2} \)
53 \( 1 - 1.34e25iT - 1.00e50T^{2} \)
59 \( 1 + 4.98e25T + 2.26e51T^{2} \)
61 \( 1 - 1.41e26iT - 5.95e51T^{2} \)
67 \( 1 - 2.07e26T + 9.04e52T^{2} \)
71 \( 1 - 4.38e24iT - 4.85e53T^{2} \)
73 \( 1 + 1.18e27iT - 1.08e54T^{2} \)
79 \( 1 + 4.92e27T + 1.07e55T^{2} \)
83 \( 1 + 1.26e28T + 4.50e55T^{2} \)
89 \( 1 - 3.53e27T + 3.40e56T^{2} \)
97 \( 1 + 6.07e28iT - 4.13e57T^{2} \)
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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

−12.27340222798639822351393492264, −11.44532005800095221518062521758, −10.43028225323859235253366354612, −8.825199898447317955402472134876, −7.66457207196150245735576248107, −6.99256432957943404815978655003, −4.47950123114619017887206508172, −4.14653802326863241531709392439, −2.49098185180482673834283026850, −1.63538571591958874355580221201, 0.39306020976481940361543374550, 1.44469162194101111725293739812, 2.94127233253238958804341196373, 3.59041060944616834704452742650, 5.51109851832810972898703198802, 7.08286744749296230883410734662, 7.991415953419920253098131790938, 8.491308509895993897477322240195, 10.88953564499712432409995217743, 11.55540939670836957018352035813

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